Getting the lowdown on points, lines, planes, and angles
Defining and combining shapes
Focusing on geometric formulas
Understanding the xy coordinate plane
Taking a closer look at triangles
The time has come for you to review America’s favorite subject, geometry. Well, it’s America’s favorite subject according to a recent survey of one person — a geometry teacher. Okay, never mind that. Geometry is basically the study of the nature of shapes and where shapes exist. The subject begins with points and goes all the way to three-dimensional figures.
Like other areas of math, geometry involves principles from areas that typically precede its study. Algebraic expressions, for example, are commonly used in geometry, and using geometric principles to solve for variables is very common. So, make sure you have a good grasp of geometry concepts because you never know when one may appear on the Praxis Core — or in real life. The math test of the Praxis exam is about 12 percent geometry. Because the math test has 56 questions, you’ll have about 7 geometry questions to answer.
Understanding the Building Blocks of Geometry
A few elements in geometry make up all the others, even the most complex geometric figures. You need to understand these elements to truly understand the more complicated figures and properties in geometry. The starting point for understanding geometric elements is the point. The starting point would automatically have to be a point. It’s not a starting prism. And then you move onto lines, planes, and angles.
Getting to the point
The most basic building block of geometry is the point, which is an exact and infinitely small location. Points make up all the physical realities studied in geometry. Points are named by capital letters that are placed near them. These are Point A and Point B:
Points aren’t always represented by dots. Dots are generally used when the location of a point would otherwise be unclear.
Defining lines and parts of lines
Any two points, anywhere in the universe, are on the same line. A line is a continuous set of an infinite number of points extending infinitely in two directions. Lines are one-dimensional.
A line can be named by any two points on it in any order, with a line symbol on top. Three of the points on the following line are labeled. The line can be called .
A line can also be named by a single italicized letter that is not a point on it. For example, the preceding line can be called Line m.
Parts of lines that serve as building blocks of geometry include rays and segments. A ray is like a line, but it has one endpoint and extends infinitely in only one direction. A ray is named by its endpoint and any other point on it, with a ray symbol on top. The following ray can be called .
A line segment, more commonly called just a segment, is a part of a line with two endpoints. Segments are named by their two endpoints, in either order. A segment symbol can be placed on top of the letters, but it isn’t necessary. This line segment is and can also be written as just PQ or QP:
You have now entered the second dimension on a plane. A plane is a flat surface that is infinitely thin and goes forever in all directions. Any two lines exist on one plane. Planes are usually named by three points that are within the plane and not all on the same line. They can also be named by a single italicized letter. The following plane can be named Plane RST, but it can also be called Plane p.
If two lines are on the same plane and never intersect, they are parallel. They remain the same distance from each other, infinitely. Parallel lines can be indicated with the same number of arrow-head marks on each line.
Getting the right angles on angles
An angle is a shape formed by two sides, with each side being either a line or part of a line. The point at which the two sides meet is called the vertex. An angle can be named by three points on it in the order of any point on either side, the vertex, and a point on the other side. You can change the order as long as the vertex stays in the middle. If the vertex is presented as the vertex of just one angle, that angle can be named by the vertex alone. The following angle can be called .
If a point is the vertex of more than one angle shown, you can’t name the angle by the vertex. That would be confusing because the angle named would not be clearly identified. The name could apply to any of the angles with that vertex. In the following illustration, which angle would be ? Nothing indicates which angle it is.
An angle named by three of its points must be named with the vertex in the middle. Otherwise, a different angle is indicated.
Recognizing congruence
Two segments or angles that have the same measure are congruent. The measure of a segment is the distance from one of its endpoints to the other, and the measure of an angle is the rate at which the sides separate as distance from the vertex increases. (We get more into angle measurement in the next section.) If two segments are congruent, they will have the same number of marks on them if such marks are used.
The congruence symbol is ≅ If AB has the same measure as CD, then AB ≅ CD. The single mark in each segment indicates that they’re congruent. The congruence of EF and GH is also indicated, but they are not congruent to AB and CD, so they have two marks each.
Angles that have the same number of arcs are congruent. In the following illustration, is congruent to , but is not congruent to either.
Angle sizes vary, so units are used to distinguish the different sizes. The unit of measure that’s most commonly used — and the only one used on the Praxis Core exam — is the degree, and it is represented by the symbol °.
Distinguishing degrees of angle measure and naming angles
An angle in which both sides are merged together could be 0 degrees, or 0°, but if one side completes a full circle, the angle it forms with the other side is 360°. An angle in which the two sides are part of the same line is 180° because one side completes half a circle with the other side.
Angles are classified according to their general sizes. An angle that is between 0° and 90° is an acute angle. If an angle is exactly 90°, it’s a right angle. An angle with a measure between 90° and 180° is an obtuse angle, and a 180° angle is a straight angle. Right angles are often symbolized by a small square near the vertex.
If two angles have measures that add up to 180°, they are supplementary. For example, a 130° angle and a 50° angle are supplementary. Because straight angles are 180°, two angles that share a side and form a straight angle together are automatically supplementary. They make what is called a linear pair because they are a pair of angles that together form a line.
Complementary angles are like supplementary angles, except two angles that are complementary have measures that add up to 90°. A 37° angle and a 53° angle are complementary. Two angles that share a side and form a right angle together are complementary.
Another major type of relationship is that of vertical angles, which are formed by two intersecting lines. Vertical angles are opposite each other, and if you have two intersecting lines, you always have two pairs of vertical angles. Vertical angles are always congruent; that is, they have the same measurement.
The correct answer is Choice (A). and form a linear pair, so they are supplementary. That means their measures have a sum of 180°. 180 – 168 is 12, so the measure of is 12°. Choice (E) is the result of adding 12 to 180. Choice (C) is the same measure as that of , Choice (D) is merely the measure of a straight angle, and Choice (B) is the difference of 270° and 168°.
When two lines are intersected by a third line at a different point each, the intersecting line is a transversal. Various angle relationships are formed in such a situation.
Angles 4 and 5, as well as angles 3 and 6, form a pair of alternate interior angles because they are on alternate sides of the transversal and interior to the given intersected lines. Angles 2 and 7 are alternate exterior angles. Corresponding angles are in the same type of position, but formed by two different lines that are intersected by the transversal. An example of a pair of corresponding angles is the pair of angles 1 and 5. Angles 4 and 6 form an example of a pair of same-side interior angles, also known as consecutive angles. They are on the same side of the transversal and interior to the lines it intersects to form the angle relationship.
If two lines intersected by a transversal are parallel, these are facts:
Each pair of alternate interior angles is congruent.
Each pair of alternate exterior angles is congruent.
Each pair of corresponding angles is congruent.
Also, if any one of those statements is a fact, it is complete proof that the lines intersected by the transversal are parallel. The rules work in both directions.
Knowing Common Shapes and Their Basic Properties
Understanding segments and the angles they form helps you analyze many of the geometric shapes you may be asked about on the Praxis Core exam. This includes the basic shapes like squares and other rectangles, triangles, and even circles because segments exist inside circles. In the following sections, we review basic shapes.
Defining polygons in general
A polygon is an enclosed figure formed on one plane by segments joined at their endpoints. Rectangles, triangles, and some other shapes are types of polygons. The polygons primarily focused on in geometry — and more particularly on the Praxis Core exam — are convex polygons, which basically are polygons that don’t point inward anywhere.
Polygons are also classified according to the number of sides they have. The number of sides a polygon has is also the number of interior angles (inside angles formed by sides that are next to each other) it has.
Analyzing triangles: Three-sided polygons
A triangle is a polygon with three sides. Because every triangle has three sides, each triangle has three interior angles also. Certain properties apply to all triangles. For example, the sum of the measures of the interior angles of a triangle is 180°. This fact is true for all triangles.
Another rule that applies to all triangles is that if two sides of a triangle are congruent, the angles opposite (across from) those sides are congruent. The rule also works in the other direction. If two angles of a triangle are congruent, the sides opposite those angles are congruent, too.
Questions on the Praxis Core exam can involve more than one rule regarding triangles and sometimes more basic rules in addition to them. You may need to use multiple rules to reach a conclusion. Many combinations of principles can be involved.
According to the diagram, what is the measure of ?
The correct answer is Choice (B). Two sides of the triangle on the left are congruent, so the angles opposite them are congruent. They are therefore both 38°. is vertical to a 38° angle, so it too is 38°. Choice (A) involves the name of the angle in question, but 1 is not given as the measure of the angle. It has no ° beside it. Choice (C) is the complement of 38°. Choice (D) is the supplement of 38°. Choice (E) results from subtracting the given angle measure from 360°. For three segments to come together, join endpoints, and form a triangle, the sum of the measures of any two segments must be greater than the measure of the third segment. Also, the third segment measure must be greater than the positive difference of the other two segment measures. The third segment has to be able to join the endpoints of the other two segments and form angles with them, so it cannot have a measure that is equal to the sum or positive difference of their measures. Very simply put, if two segment measures are given, the third segment measure has to be between (not equal to) the sum and the positive difference of the other two segment measures. Imagine trying to form a triangle with these three segments.
It would be impossible. The two shorter segments could not join together, form an angle with a measure between zero and 180 degrees, and reach the endpoints of the third segment. The sum of the measures of the two shortest segments is less than the measure of the longest segment. Also, the longest segment measure minus either one of the other segment measures is greater than the remaining segment measure. There is no way for a segment measure to be between the sum and positive difference of the other two segment measures. That makes forming a triangle with the segments impossible.
If you are given two segment measures and asked within what range the third segment measure must be, you can find the sum and difference of the two given measures. The third must be within those. If you are given three segment measures and asked whether the segments can join at endpoints to form a triangle, the shortcut to the answer is to take the two shortest segments and see whether the sum of their measures is greater than the longest. If their sum is greater, then the segments can form a triangle. That is because if the sum of the two shortest segment measures is greater than the third segment measure, the other sums of two segment measures will have to be greater than the third. If it works with the two shortest, it has to work with the other combinations. Plus, the longest segment measure cannot possibly be less than the positive difference of the two shortest segment measures. That one test covers all bases. (No pun intended.)
Consider a 3 m segment, a 9 m segment, and a 7 m segment. Can they form a triangle? The two shortest segments are the 3 m and 7 m segments. The sum of those measures is 10 m, which is greater than 9 m. That means the segments can form a triangle. Since the segments pass that test, they pass the other tests.
As triangles are three-sided polygons, quadrilaterals are four-sided polygons. Squares and other rectangles are quadrilaterals, but they are not the only types. The sum of the interior angles of a quadrilateral is always 360°.
A quadrilateral in which both pairs of opposite sides are parallel is a parallelogram. For any parallelogram, both pairs of opposite sides are also congruent, and both pairs of opposite angles are congruent. Also, any quadrilateral in which both pairs of opposite sides are congruent is a parallelogram, and any quadrilateral in which both pairs of opposite angles are congruent is a parallelogram. A rectangle is a quadrilateral in which all four interior angles are right angles, and all rectangles are parallelograms, so their opposite sides are congruent. The diagonals of a rectangle are also congruent. A square is a rectangle in which all four sides are congruent.
A trapezoid is a quadrilateral that has just one pair of parallel sides. The two parallel sides are the bases of the trapezoid. The bases in the following trapezoid are indicated by arrows, which are used in geometry to suggest that lines and parts of lines are parallel.
No limit exists for the number of sides a polygon can have. Many names are used for types of polygons based on the number of sides they have. For example, a five-sided polygon is a pentagon, and an eight-sided polygon is an octagon. However, the Praxis Core exam doesn’t focus on the major rules concerning polygons that have more than four sides. What you need to be able to do is recognize what is inside such polygons. Rules you need to know may apply to segments, angles, triangles, quadrilaterals, and other formations that are within them.
According to the diagram, what is the measure of ?
The correct answer is Choice (E). The pentagon is divided into a quadrilateral and a triangle. Because the interior angle sum of a quadrilateral is 360°, the interior angle without a given measure in the quadrilateral is 63°. That angle is complementary to an angle in the triangle, so that angle in the triangle has a measure of 27°. Because another angle in the triangle is 100° and the interior angle sum of a triangle is 180°, has to be 53°. The other choices can result from the wrong uses of formulas.
Knowing the basic facts about circles
A circle is a shape like the others covered so far, but a circle has no sides. A circle is the set of all points in one plane that are a given distance from a point called the center. The distance that all the points are from the center is the radius of the circle. A radius is also an actual segment that connects the center to a point on the circle. The diameter of a circle is the distance across the interior through the center, and it is also the name of a segment that covers the path. The diameter of a circle is always twice the radius. Because the radius of the following circle is 3 centimeters, the diameter is 6 centimeters. A circle is named by its center, so this circle is Circle K.
Polygons can be alike in certain ways, and so can circles. Polygons that have the same number of sides don’t necessarily have the same shape, but they do if they are congruent or similar.
Forming conclusions about congruent shapes
When two polygons have the same number of sides and all of their corresponding (same position, different polygons) pairs of sides and corresponding pairs of angles are congruent, the polygons themselves are congruent. Also, if you are given the fact that two polygons are congruent, you have enough information to conclude that every pair of corresponding parts is congruent.
The two following triangles are congruent. Notice that every side and angle of one triangle is congruent to its corresponding part in the other triangle. They are the same triangle in two different places, exact copies of each other. As a result, all of their corresponding parts are congruent.
All other types of polygons can be congruent. Circles can be congruent too. If two circles have the same radius, they are congruent. They have the same diameter and other measures. We get into circumference and area in a bit, but for now it’s enough to say that congruent circles have the same circumference (distance around the circle) and the same area. The following two circles are congruent.
Similar shapes are exactly the same shape but not necessarily congruent. Imagine magnifying or reducing a picture of a quadrilateral. The resulting image would look identical to the original except that it would be a different size. The two shapes would be similar.
Polygons that are similar have congruent corresponding angles. They also have side measures that are proportional, which means that the ratio of the measure of a side in one polygon to the measure of the side that corresponds to it in the other polygon is always the same ratio. Recall that a proportion is an equation in which one ratio is equal to another. The ratio of one side measure to its corresponding side measure is the scale factor of the similarity relationship. Scale factor depends on which figure is put first in the ratio, so two scale factors can exist between two figures. For the following two triangles, the scale factor is 2:1 or 1:2, which can also be written as 2/1 or 1/2.
The way to determine the measure of a side of a polygon that’s similar to another polygon is to compare ratios of corresponding sides within the two polygons. You can do that by writing a side measure over its corresponding side measure and setting it equal to another such ratio. Make sure you are consistent in the way you write the ratios. If you put the first polygon measure on top in one ratio, the first polygon measure has to go on top in the other ratio.
Suppose the triangle side that is 7 feet was of unknown measure, but the other measures were given. You could find the side measure by letting a variable represent it and solving a proportion involving it.
Corresponding angles of similar polygons are congruent.
You can apply these principles of shape similarity to maps of geographical regions, floor plans of houses, and other representations of places in reality. Such representations are called scale models, and they are shaped like the places they represent. You can use similarity principles to determine distances in real-world places based on distances in the scale models that represent the places. Finding the answers to such problems generally requires use of a given scale factor, usually presented in a sample ratio.
Figuring out Geometric Formulas
Geometric figures have certain properties, and the number of dimensions they have is part of what decides what properties they have. Line segments have a distance that can be referred to as length, width, or height. Two-dimensional figures such as circles and triangles have area as well as parts with one-dimensional measurements. Three-dimensional geometric figures have the preceding properties plus volume.
Formulas are provided on the Praxis Core, but you will need to know what the variables in them represent. You should also practice using the formulas before you take the exam to make sure you know what you are doing and to help avoid errors.
Finding the perimeter
The perimeter of a two-dimensional figure is the distance around it. To determine the perimeter of a polygon, you can add all the side measures. The following rectangle has a perimeter of 28 meters.
Because the opposite sides of a rectangle are congruent, a formula makes calculating the perimeter simpler than adding up all the side measures. Two of the sides have the length (l), and two sides have the width (w), so adding twice the length and twice the width gives the perimeter:
Perimeter of rectangle = 2l + 2w
Because the length and width of a square are the same, you can get the perimeter by multiplying the measure of one side by 4.
Circling the circumference
The perimeter of a circle is the circle’s circumference. The formula for circumference involves π, which is the ratio of a circle’s circumference (C) divided by its diameter (d). Because all circles are similar, the ratio is the same for all circles.
π is an irrational number, so it never terminates or repeats in decimal form, but its value can be rounded to 3.14159. Because circumference divided by diameter is π, circumference is diameter times π:
The diameter of a circle is twice the radius, so . Therefore, . The formal way to write a term is with numbers before variables, and π is a number, so the official formula for the circumference is this:
Remember that within a formula, any variable can represent an unknown in a problem. To find the value of the variable, fill every known number into the formula and solve for what is not yet known.
What is the radius of a circle with a circumference of 10π units?
(A) 10
(B) 5
(C) 100
(D) 5π
(E) 10π
The correct answer is Choice (B). You can use the formula for circumference and solve for r.
The other choices result from misuse of the circumference formula or using the wrong formula.
If a question involves π, the symbol π may or may not appear as part of the answer choices. If it is not part of the answer choices, then it has probably been calculated in approximated decimal form. In that case, you need to use 3.14159, the approximation for π, and do the calculation. The number 3.14159 is not exactly π, but you can use it to get really close approximations.
Getting into the area
A two-dimensional figure exists on a plane. The area of a two-dimensional figure is the amount of plane in it. In other words, area is a measure of how much room is inside a two-dimensional shape. Different shapes have different area formulas.
The area of a parallelogram is its base times its height. The base can be any side, but the height has to be the measure of a segment that is perpendicular to it and its opposite side.
area of parallelogram = bh
The area of the following parallelogram is its base times its height, or (7 cm)(10 cm), or 70 cm2.
Any combination of base and height for a rectangle, which is a type of parallelogram, is a combination of length (l) and width (w), so the area of a rectangle is lw. Length and width are the same for a square because all four sides are congruent, so to get the area of a square, all you have to do is multiply a side measure by itself. In other words, square a side measure.
If a parallelogram is cut at the vertices, the result is two congruent triangles. Also, any triangle can be put together with a congruent triangle to form a parallelogram. Because of this, every triangle has half the area of the parallelogram that can be formed by putting the triangle with an exact copy of itself. Therefore, the area of a triangle is not base times height, but half that:
The area of the following triangle , or , which is 44 ft2.
Table 6-1 shows the formulas for finding the areas of common shapes. Make sure you are very familiar with these because you’ll probably be asked at least one question about area on the Praxis.
The Praxis Core may ask you to make a calculation concerning the volume of a rectangular prism, which is a three-dimensional figure made up of six rectangles joined together by their sides. Each rectangle is a face. All of the angles of a rectangular prism are right angles, and all pairs of faces that are across from each other are parallel. A rectangular prism is basically a box.
Volume is a three-dimensional measure. It is the amount of space inside a three-dimensional figure. For rectangular prisms, the volume can be found by multiplying the base area by the height. The volume is more specifically lwh because lw is the base area. Any face can be considered a base, but the height must be the measure of a side that connects the face considered the base to the face across from it. You can measure volume in cubic units, or units to the third power. For example, if the length, width, and height of a rectangular prism are presented on the exam in cm, the volume will be presented in .
Rectangular prism volume = lwh
A cube is a type of rectangular prism in which all six faces are squares that are congruent to each other. For any cube, the length, width, and height are equal, so the volume is equal to the cube of a side measure.
Combining Shapes
The Praxis Core exam may ask you about areas or volumes of figures created by joining figures of the types covered so far in this chapter. In some situations, the mergers of figures add area or volume, and in others, you may be asked for the area or volume of a shaded region that exists outside of another figure, in which cases area or volume is reduced.
Coming together: Shapes that have joined without an invasion
When shapes join without one intruding on the other, you can find the total area or total volume of the figure they form together by adding the areas or volumes of the figures that form it. For example, a triangle may share a side with a square, or a cone may share a base with a cylinder. Just add the areas or volumes of the figures to find the total area or volume.
The figure shown has a height of 14 cm and is composed of a square and a triangle that share a side. What is the area of the figure that they form?
The correct answer is Choice (A). One side of the square is 6 cm, so all of its sides are 6 cm. That means the base of the triangle is also 6 cm. Because the height of the overall figure is 14 cm and a side of the square is 6 cm, the height of the triangle is 8 cm. The area of the square is s2, or 36 cm2, and the area of the triangle is , or , which is 24 cm2. The sum of the areas of the square and the triangle is 36 cm2 + 24 cm2, or 60 cm2. The other choices can result from incorrect calculations involving measures in the diagram.
Preparing for invasion: When one shape invades another
If a question on the Praxis Core exam involves a shape that exists within another one and you’re asked to find the area or volume of the shape region that is outside of the intruding shape, subtract the area or volume of the intruding shape from that of the shape upon which it is intruding.
To find the area of the shaded region of the rectangle that follows, you can get the area of the rectangle and then subtract the area of the triangle. The remaining area is the area of the shaded region. The area of the rectangle is (12 mi)(5 mi), or 60 mi2. The height of the triangle is the same as the width of the rectangle, so the area of the triangle is , or 30 mi2. 60 mi2 – 30 mi2 = 30 mi2, so the remaining shaded region has an area of 30 mi2.
The coordinate plane is a two-dimensional system of points that are named by their positions in regard to two intersecting number lines, the x-axis and the y-axis. The x-axis is horizontal, and the y-axis is vertical. Using the xy coordinate plane, you can find and name the locations of points, lines, parts of lines, graphs of equations and inequalities, and two-dimensional figures.
Naming coordinate pairs
Every point on the coordinate plane is named by two numbers, the first of which is an x-coordinate, which indicates a point’s position along the x-axis, and the second of which is a y-coordinate, an indication of a point’s position along the y-axis. The point of intersection of the x and y axes is called the origin, and its coordinates are (0, 0).
To determine the coordinates of a point, first locate the origin. Then determine which number on the x-axis the point is on or directly above or below. That is the x-coordinate of the point. Then determine the number on the y-axis the point is on or next to; that is the y-coordinate. To think about it another way, the number of horizontal units you move from the origin is the x-coordinate, and the number of vertical units you move from the origin is the y-coordinate. Several points and their corresponding ordered pairs are shown on the following coordinate plane.
Up and right are positive directions on the coordinate plane, and down and left are negative directions.
Identifying linear equations and their graphs
Every line on the coordinate plane is the graph of a linear equation. Every point on a line represents an ordered pair that makes a linear equation true. In other words, if the x-coordinate is put in for x in the equation and the y value is put in for y, the equation is true. The graph therefore represents all the ordered pairs that make the equation work or that are solutions to the equation.
Not all equations are linear. Equations that are linear have certain characteristics. The variables x and y are generally used in linear equations. Also, some linear equations have only one variable. At least one variable is necessary for the equation to be linear. Also, no variable has an exponent other than 1, which is generally understood and not shown. The variables are never exponents or multiplied by each other in linear equations. For example, , , and are all linear equations. The following graph represents . Notice how every point on the graph represents an ordered pair that makes the equation true.
The slope of a line is an indication of the line’s steepness and direction. It is a ratio of the rate of change in y to the rate of change in x. With any two points on a line on the coordinate plane, you can find the slope of the line. The change in y is the difference of the y-coordinates, and the change in x is the difference of the x-coordinates.
In the formula for slope, (x1, y1) is one point and (x2, y2) represents the other point. Which point you call which doesn’t matter, but make sure you are consistent. In other words, subtract in the same direction both times.
Another way to determine the slope of a line is to write the equation in slope-intercept form, , where m represents slope and b represents y-intercept, which is the y-coordinate of the point where the line intersects the y-axis. For example, the equation can be rewritten as , which shows that the slope of the graph is 4 because it is the coefficient of x when the equation is in slope-intercept form. Just get y by itself on the left side and put the x term first on the right side. (If you need a review of how to isolate variables, flip back to Chapter 5.) Once the equation is in that form, the coefficient of x is the slope of the graph of the line.
To determine whether a graph represents an equation, you can test the points on the graph. Every ordered pair represented by the graph must make the equation true. For a linear equation, testing two points is sufficient because only one line can pass through two points.
Using the distance and midpoint formulas
The distance between two points on the coordinate plane is given by the formula for distance . The formula is based on x and y changes and the Pythagorean theorem, which we cover in the section “Knowing what Pythagoras discovered.” The distance between the points (5, 7) and (9, 4) is
The distance between the points is 5.
The midpoint between two points is the point halfway between them. Its x-coordinate is halfway between the two points’ x-coordinates, and its y-coordinate is halfway between the two points’ y-coordinates. The number that is halfway between two other numbers is the average of the two numbers. Thus, the midpoint between two points is the average of the x-coordinates followed by a comma and then the average of the y-coordinates.
Midpoint:
The midpoint between (–2, 7) and (4, 3) is
The distance between two points is a single number, while the midpoint between two points is a point, represented by an ordered pair.
Finding meaning in intersecting graphs
Because every point on the graph of an equation represents an ordered pair that is a solution to the equation, a point on two different equation graphs is a solution to both equations. Therefore, when two graphs intersect, their point of intersection represents a solution to both graphs’ equations. The following graph represents the equations and . They intersect at (5, 2), so (5, 2) is the solution to both equations. In other words, if you put 5 in for x and 2 in for y in either equation, you get an equation that is true.
The figures can be altered in various ways by changing the locations of their vertices. Such changes are called transformations. Four major types of transformations can be covered on the Praxis Core exam. One type of transformation is a translation, which involves simply moving the figure from one location to another. Translating a figure involves sliding it a number of units horizontally and a number of units vertically. The number can, of course, be 0 for one of the changes.
In the following graph, a triangle has vertices (–2, –3), (0, 0), and (4, –5). If it is translated 3 units left and 2 units up, 3 is subtracted from each x-coordinate and 2 is added to each y-coordinate to get the coordinates of the vertices of the new triangle. Thus, the new vertex coordinates are (–5, –1), (–3, 2), and (1, –3).
Reflections involve flipping figures over an axis and creating new images that are like mirror reflections. To reflect a figure over the x-axis, change the y-coordinates of its points to their opposites. If the figure is a polygon, changing the vertices is enough. To reflect a figure over the y-axis, change the x-coordinates of the points to their opposites to get the new points. The following graph shows a reflection of a quadrilateral over the y-axis.
Dilations are simply changes in the sizes of figures. To dilate a figure by a certain amount, multiply both coordinates of the points by the same number. The triangle in the graph that follows is dilated to three times its original size. This is achieved by multiplying all of its vertex coordinates by 3 to get the new coordinates.
Rotations involve the moving of figures along circular paths while the distance between every point and the center of the circular path stays the same. The types of rotations you are likely to see on the Praxis Core exam are 180° and 90° counterclockwise around the origin. To rotate a figure 180° around the origin, get the opposite of each coordinate. Those will be the new coordinates. To rotate 90° around the origin, switch the coordinates of each point and get the opposite of the resulting x-coordinate. In the following figure, a line segment is rotated 90° and 180° about the origin.
You can expect to see some questions about right triangles on the Praxis Core exam. A right triangle is a triangle with one right angle. A triangle can have no more than one right angle because the sum of the angles is 180°, and two right angles meet that number, not allowing for a third angle. The two sides of a right triangle that form the right angle are the legs, and the side across from the right angle is the hypotenuse.
Pythagoras was a Greek philosopher and mathematician who discovered a principle that became known as the Pythagorean theorem, which states that the sum of the squares of the two leg measures of a right triangle is equal to the square of the measure of the hypotenuse. The theorem is often represented by the equation , where a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse.
Questions on the Praxis Core exam can ask for the measure of the hypotenuse when both leg measures are given, but they can also involve the measures of the hypotenuse and one leg being given when you are asked for the measure of the other leg. To determine the answer to any such question, use the formula , fill in the known values, and solve for the unknown value. Really, that’s how all formulas should be used.
If a leg of a right triangle is 3 cm and the hypotenuse is 5 cm, what is the measure of the other leg?
(A) 16 cm
(B) 8 cm
(C) 2 cm
(D) 4 cm
(E) 15 cm
The correct answer is Choice (D). Take the information you know, plug it into the formula, and solve for the information the question asks for:
If a geometric figure is described but not illustrated, draw the figure so you have a visual representation. That will make working the problem easier.
Working with special right triangles
The Praxis Core exam may ask you a question that involves one of two types of special right triangles — a 45-45-90 or a 30-60-90 triangle. Each is named after the combination of angle degree measures. They both sound like football cheers, but that wasn’t the original idea. Both types have special rules concerning side measures because every example of each is similar. Recall that when the corresponding angles of two triangles are congruent, the triangles are similar.
All 45-45-90 triangles are isosceles, which means they have two congruent sides. A 60-60-60 triangle is equilateral, which means all three of its sides are congruent. All 45-45-90 triangles are similar, all 30-60-90 triangles are similar, and all 60-60-60 triangles are similar. However, 60-60-60 triangles are not right triangles. A right triangle has one 90° angle.
For every 45-45-90 triangle, the two legs are congruent because the angles opposite them are congruent. The hypotenuse is always times the measure of a leg.
For every 30-60-90 triangle, the hypotenuse measure is twice that of the shorter leg (which is opposite the 30° angle), and the longer leg is times the measure of the shorter leg.
3. What is the circumference of a circle that has an area of 25π cm2?
(A) 5 cm
(B) 12.5 cm
(C) 12.5π cm
(D) 10π cm
(E) 10 cm
4. The volume of a rectangular prism is 60 m3. The length of a base is 5 m, and the width of that base is 4 m. What is the accompanying height of the rectangular prism?
(A) 1 m
(B) 3 m
(C) 5 m
(D) 7 m
(E) 11 m
5. A segment on the coordinate plane has endpoints (8, 6) and (4, 2). If a line intersects the segment at (6, 4), which of the following statements is true?
Indicate all such statements.
(A) The point of intersection is closer to (8, 6) than (4, 2).
(B) The point of intersection is closer to (4, 2) than (8, 6).
(C) The point of intersection is equally close to the endpoints of the segment.
(D) The point of intersection is the midpoint of the segment.
(E) The proximity of the point of intersection to the endpoints of the segment cannot be determined.
Answers and Explanations
Use this answer key to score the practice geometry questions in this chapter.
A. The hypotenuse is 12, and the shorter leg is half that measure, or 6. The longer leg is the shorter leg measure times , so the measure of the longer leg is . Choices (C) and (E) involve the use of , which is part of a formula for the measure of the hypotenuse of a 45-45-90 triangle. Be careful. Choices (B) and (D) can result from careless errors involving the given information.
D. 44°. The triangle that has an angle measure of x° has an angle that is a vertical angle to an 86° angle, and vertical angles are congruent. The third angle of the same triangle is supplementary to a 130° angle because they form a linear pair, so that third angle is 50°. The sum of 86 and 50 is 136. The interior angle sum of a triangle is 180°, and 180° – 136° = 44°. Therefore, x is 44. The other choices can result from using false principles involving the measurements in the diagram.
D. 10π cm. To solve this problem, you must use what you know to find the value of the radius, which is needed for the circumference formula. Because the area of a circle is πr2, . That means . The circumference of a circle is 2πr. So . The other choices can result from misuse of operations involved in the necessary steps.
B. 3 m. The volume of a rectangular prism is equal to a base area times the height that goes with the given base. In short, the volume is length × width × height, or lwh. For this problem, l and w are given. You can plug their values into the formula and solve for the unknown, h.
The height is 3 m.
C. The point of intersection is equally close to the endpoints of the segment. D. The point of intersection is the midpoint of the segment. The best place to start is getting the midpoint between the endpoints of the segment. That point is the midpoint of the segment. Determining it will help you figure out where the point of intersection is in relation to the endpoints of the segment. The midpoint between two points is , or the average of the x-coordinates and the average of the y-coordinates separated by a comma. The average of 8 and 4 is 6, and the average of 6 and 2 is 4, so the midpoint is (6, 4). That is also the point of intersection, so the point of intersection is the midpoint of the segment and is therefore equally close to the endpoints of the segment.
Getting the lowdown on points, lines, planes, and angles
. 
. 
and can also be written as just PQ or QP: 
. 
If a point is the vertex of more than one angle shown, you can’t name the angle by the vertex. That would be confusing because the angle named would not be clearly identified. The name could apply to any of the angles with that vertex. In the following illustration, which angle would be 
? Nothing indicates which angle it is. 
An angle named by three of its points must be named with the vertex in the middle. Otherwise, a different angle is indicated.
is congruent to 
, but 
is not congruent to either. 
The measure of 
is 168°. What is the measure of 
? 
and 
form a linear pair, so they are supplementary. That means their measures have a sum of 180°. 180 – 168 is 12, so the measure of 
is 12°. Choice (E) is the result of adding 12 to 180. Choice (C) is the same measure as that of 
, Choice (D) is merely the measure of a straight angle, and Choice (B) is the difference of 270° and 168°. 
? 
is vertical to a 38° angle, so it too is 38°. Choice (A) involves the name of the angle in question, but 1 is not given as the measure of the angle. It has no ° beside it. Choice (C) is the complement of 38°. Choice (D) is the supplement of 38°. Choice (E) results from subtracting the given angle measure from 360°. For three segments to come together, join endpoints, and form a triangle, the sum of the measures of any two segments must be greater than the measure of the third segment. Also, the third segment measure must be greater than the positive difference of the other two segment measures. The third segment has to be able to join the endpoints of the other two segments and form angles with them, so it cannot have a measure that is equal to the sum or positive difference of their measures. Very simply put, if two segment measures are given, the third segment measure has to be between (not equal to) the sum and the positive difference of the other two segment measures. Imagine trying to form a triangle with these three segments. 
? 
has to be 53°. The other choices can result from the wrong uses of formulas. 
Because the length and width of a square are the same, you can get the perimeter by multiplying the measure of one side by 4.
. Therefore, 
. The formal way to write a term is with numbers before variables, and π is a number, so the official formula for the circumference is this: 
, or 
, which is 44 ft2. 
.
, or 
, which is 24 cm2. The sum of the areas of the square and the triangle is 36 cm2 + 24 cm2, or 60 cm2. The other choices can result from incorrect calculations involving measures in the diagram. 
, or 30 mi2. 60 mi2 – 30 mi2 = 30 mi2, so the remaining shaded region has an area of 30 mi2. 
, 
, and 
are all linear equations. The following graph represents 
. Notice how every point on the graph represents an ordered pair that makes the equation true. 
, where m represents slope and b represents y-intercept, which is the y-coordinate of the point where the line intersects the y-axis. For example, the equation 
can be rewritten as 
, which shows that the slope of the graph is 4 because it is the coefficient of x when the equation is in slope-intercept form. Just get y by itself on the left side and put the x term first on the right side. (If you need a review of how to isolate variables, flip back to 
. The formula is based on x and y changes and the Pythagorean theorem, which we cover in the section “
and 
. They intersect at (5, 2), so (5, 2) is the solution to both equations. In other words, if you put 5 in for x and 2 in for y in either equation, you get an equation that is true. 
, where a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse.
, fill in the known values, and solve for the unknown value. Really, that’s how all formulas should be used.
times the measure of a leg. 
times the measure of the shorter leg. 
meters
meters
meters
The hypotenuse is 12, and the shorter leg is half that measure, or 6. The longer leg is the shorter leg measure times 
, so the measure of the longer leg is 
. Choices (C) and (E) involve the use of 
, which is part of a formula for the measure of the hypotenuse of a 45-45-90 triangle. Be careful. Choices (B) and (D) can result from careless errors involving the given information.
. That means 
. The circumference of a circle is 2πr. So 
. The other choices can result from misuse of operations involved in the necessary steps.
, or the average of the x-coordinates and the average of the y-coordinates separated by a comma. The average of 8 and 4 is 6, and the average of 6 and 2 is 4, so the midpoint is (6, 4). That is also the point of intersection, so the point of intersection is the midpoint of the segment and is therefore equally close to the endpoints of the segment.