Constructions

15.1   Introduction

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Geometric figures are constructed with straightedge and compass. Since constructions are based on deductive reasoning, measuring instruments such as the ruler and protractor are not permitted. However, a ruler may be used as a straightedge if its markings are disregarded.

In constructions, it is advisable to plan ahead by making a sketch of the situation; such a sketch will usually reveal the needed construction steps. Construction lines should be made light to distinguish them from the required figure.

The following constructions are detailed in this chapter:

1.  To construct a line segment congruent to a given line segment

2.  To construct an angle congruent to a given angle

3.  To bisect a given angle

4.  To construct a line perpendicular to a given line through a given point on the line

5.  To bisect a given line segment

6.  To construct a line perpendicular to a given line through a given external point

7.  To construct a triangle given its three sides

8.  To construct an angle of measure 60°

9.  To construct a triangle given two sides and the included angle

10.  To construct a triangle given two angles and the included side

11.  To construct a triangle given two angles and a side not included

12.  To construct a right triangle given its hypotenuse and a leg

13.  To construct a line parallel to a given line through a given external point

14.  To construct a tangent to a given circle through a given point on the circle

15.  To construct a tangent to a given circle through a given point outside the circle

16.  To circumscribe a circle about a triangle

17.  To locate the center of a given circle

18.  To inscribe a circle in a given triangle

19.  To inscribe a square in a given circle

20.  To inscribe a regular octagon in a given circle

21.  To inscribe a regular hexagon in a given circle

22.  To inscribe an equilateral triangle in a given circle

23.  To construct a triangle similar to a given triangle on a given line segment as base

15.2   Duplicating Segments and Angles

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CONSTRUCTION 1:     To construct a line segment congruent to a given line segment

Given:   Line segment (Fig. 15-1)

To construct:   A line segment congruent to

Construction:   On a working line w, with any point C as a center and a radius equal to AB, construct an arc intersecting w at D. Then is the required line segment.

Fig. 15-1

Fig. 15-2

CONSTRUCTION 2:     To construct an angle congruent to a given angle

Given:    (Fig. 15-2)

To construct:   An angle congruent to

Construction:   With A as center and a convenient radius, construct an arc (1) intersecting the sides of at B and C. With A′, a point on a working line w, as center and the same radius, construct arc (2) intersecting w at B′. With B′ as center and a radius equal to BC, construct arc (3) intersecting arc (2) at C′. Draw A′C′. Then is the required angle. ( by SSS; hence .)

SOLVED PROBLEMS

15.1   Combining line segments

Given line segments with lengths a and b (Fig. 15-3), construct line segments with lengths equal to (a) a + 2b; (b) 2(a + b); (c) b – a.

Fig. 15-3

Solutions

Use construction 1.

(a)  On a working line w, construct a line segment with length a. From B, construct a line segment with length equal to b, to point C; and from C construct a line segment with length b, to point D. Then is the required line segment.

(b)  Similar to (a). AD = a + b + (a + b).

(c)  Similar to (a). First construct with length b, then with length a. AC = b – a.

15.2   Combining angles

Given in Fig. 15-4, construct angles whose measures are equal to (a) 2A; (b) A + B + C; (c) B – A.

Fig. 15-4

Solutions

Use construction 2.

(a)  Using a working line w as one side, duplicate . Construct another duplicate of adjacent to , as shown. The exterior sides of the copied angles form the required angle.

(b)  Using a working line w as one side, duplicate . Construct adjacent to . Then construct adjacent to . The exterior sides of the copied angles A and C form the required angle. Note that the angle is a straight angle.

(c)  Using a working line w as one side, duplicate . Then duplicate from the new side of as shown. The difference is the required angle.

15.3   Constructing Bisectors and Perpendiculars

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CONSTRUCTION 3:     To bisect a given angle

Given:    (Fig. 15-5)

To construct:   The bisector of

Construction:   With A as center and a convenient radius, construct an arc intersecting the sides of at B and C. With B and C as centers and equal radii, construct arcs intersecting in D. Draw . Then is the required bisector. ( by SSS; hence, .)

Fig. 15-5

Fig. 15-6

CONSTRUCTION 4:     To construct a line perpendicular to a given line through a given point on the line

Given:   Line w and point P on w (Fig. 15-6)

To construct:   A perpendicular to w at P

Construction:   Using construction 3, bisect the straight angle at P. Then is the required perpendicular; is the required line.

CONSTRUCTION 5:     To bisect a given line segment (to construct the perpendicular bisector of a given line segment)

Given:   Line segment (Fig. 15-7)

To construct:   The perpendicular bisector of

Construction:   With A as center and a radius of more than half , construct arc (1). With B as center and the same radius, construct arc (2) intersecting arc (1) at C and D. Draw . is the required perpendicular bisector of . (Two points each equidistant from the ends of a segment determine the perpendicular bisector of the segment.)

Fig. 15-7

Fig. 15-8

CONSTRUCTION 6:     To construct a line perpendicular to a given line through a given external point

Given:   Line w and point P outside of w (Fig. 15-8)

To construct:   A perpendicular to w through P

Construction:   With P as center and a sufficiently long radius, construct an arc intersecting w at B and C. With B and C as centers and equal radii of more than half , construct arcs intersecting at A. Draw . Then is the required perpendicular. (Points P and A are each equidistant from B and C.)

SOLVED PROBLEMS

15.3   Constructing special lines in a triangle

In scalene [Fig. 15-9(a)], construct (a) a perpendicular bisector of and (b) a median to . In [Fig. 15-9(b)], D is an obtuse angle; construct (c) the altitude to and (d) the bisector of .

Fig. 15-9

Solutions

(a)  Use construction 5 to obtain the perpendicular bisector of .

(b)  Point M is the midpoint of . Draw , the median to .

(c)  Use construction 6 to obtain , the altitude to (extended).

(d)  Use construction 3 to bisect . is the required bisector.

15.4   Constructing bisectors and perpendiculars to obtain required angles

(a)  Construct angles measuring 90°, 45°, and 135°.

(b)  Given an angle with measure A (Fig. 15-10), construct an angle whose measure is 90° + A.

Fig. 15-10

Solutions

(a)  In Fig. 15-10(a), = 90°, = 45°, = 135°

(b)  In Fig. 15-10(b), = 90° + A.

15.4   Constructing a Triangle

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15.4A   Determining a Triangle

A triangle is determined when a set of given data fix its size and shape. Since the parts needed to prove congruent triangles fix the size and shape of the triangles, a triangle is determined when the given data consist of three sides, or two sides and the angle included by those sides, or two angles and a side included by those angles, or two angles and a side not included by those angles, or the hypotenuse and either leg of a right triangle.

15.4B   Sketching Triangles to be Constructed

Before doing the actual construction, it is very helpful to make a preliminary sketch of the required triangle. In this sketch:

1.   Show the position of each of the given parts of the triangle.

2.   Draw the given parts heavy, the remaining parts light.

3.   Approximate the sizes of the given parts.

4.   Use small letters for sides to agree with the capital letters for the angles opposite them.

Fig. 15-11

As an example, you might make a sketch like that in Fig. 15-11 before constructing a triangle given two angles and an included side.

15.4C   Triangle Constructions

CONSTRUCTION 7:     To construct a triangle given its three sides

Given:   Sides of lengths a, b, and c (Fig. 15-12)

To construct:   

Construction:   On a working line w, construct such that AC = b. With A as center and c as radius, construct arc (1). Then with C as center and a as radius, construct arc (2) intersecting arc (1) at B. Draw and . is the required triangle.

Fig. 15-12

Fig. 15-13

CONSTRUCTION 8:     To construct an angle of measure 60°

Given:   Line w (Fig. 15-13)

To construct:   An angle of measure 60°

Construction:   Using a convenient length as a side, construct an equilateral triangle using construction 7. Then any angle of the equilateral triangle is the required angle.

CONSTRUCTION 9:     To construct a triangle given two sides and the included angle

Given:   , segments of lengths b and c (Fig. 15-14)

To construct:   

Construction:   On a working line w, construct such that AC = b. At A, construct with one side . On the other side of , construct such that AB = c. Draw . Then the required triangle is .

Fig. 15-14

CONSTRUCTION 10:   To construct a triangle given two angles and the included side

Given:   , , and a segment of length b (Fig. 15-15)

To construct:   

Construction:   On a working line w, construct such that AC = b. At A, construct with one side on , and at C, construct with one side on . Extend the new sides of the angles until they meet at B.

Fig. 15-15

CONSTRUCTION 11:   To construct a triangle given two angles and a side not included

Given:   , , and a segment of length b (Fig. 15-16)

To construct:   

Construction:   On a working line w, construct such that AC = b. At C, construct an angle with measure equal to + so that the extension of will be one side of the angle. The remainder of the straight angle at C will be . At A, construct with one side on . The intersection of the new sides of the angles is B.

Fig. 15-16

Fig. 15-17

CONSTRUCTION 12:   To construct a right triangle given its hypotenuse and a leg

Given:   Hypotenuse with length c and leg with length b of right triangle ABC (Fig. 15-17)

To construct:   Right triangle ABC

Construction:   On a working line w, construct such that AC = b. At C construct a perpendicular to . With A as center and a radius of c, construct an arc intersecting the perpendicular at B.

SOLVED PROBLEMS

15.5   Constructing a triangle

Construct an isosceles triangle, given the lengths of the base and an arm (Fig. 15-18).

Fig. 15-18

Solution

Use construction 7, since all three sides of the triangle are known.

15.6   Constructing angles based on the construction of the 60° angle

Construct an angle of measure (a) 120°; (b) 30°; (c) 150°; (d) 105°; (e) 75°.

Solutions

(a)  Use construction 8 [Fig. 15-19(a)] to construct 120° as 180°–60°.

(b)  Use constructions 8 and 3 to construct 30° as (60°) [Fig. 15-19(b)].

(c)  Use (b) to construct 150° as 180°–30° [Fig. 15-19(b)].

(d)  Use constructions 3, 4, and 8 to construct 105° as 60° + (90°) [Fig. 15-19(c)].

(e)  Use (d) to construct 75° as 180° – 105° [Fig. 15-19(c)].

Fig. 15-19

15.5   Constructing Parallel Lines

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CONSTRUCTION 13:   To construct a line parallel to a given line through a given external point

Given:    and external point P (Fig. 15-20)

To construct:   A line through P parallel to

Construction:   Draw a line through P intersecting in Q. Construct . Then is the required parallel. (If two corresponding angles are congruent, the lines cut by the transversal are parallel.)

Fig. 15-20

SOLVED PROBLEM

15.7   Constructing a parallelogram

Construct a parallelogram given the lengths of two adjacent sides a and b and of a diagonal d (Fig. 15-21).

Fig. 15-21

Solution

Three vertices of the parallelogram are obtained by constructing by construction 7. The fourth vertex, C, is obtained by constructing upon diagonal by construction 7. Vertex C may also be obtained by constructing and .

15.6   Circle constructions

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CONSTRUCTION 14:   To construct a tangent to a given circle through a given point on the circle

Given:   Circle O and point P on the circle (Fig. 15-22)

To construct:   A tangent to circle O at P

Construction:   Draw radius and extend it outside the circle. Construct is the required tangent. (A line perpendicular to a radius at its outer extremity is a tangent to the circle.)

Fig. 15-22

Fig. 15-23

CONSTRUCTION 15:   To construct a tangent to a given circle through a given point outside the circle

Given:   Circle O and point P outside the circle (Fig. 15-23)

To construct:   A tangent to circle O from P

Construction:   Draw , and make the diameter of a new circle Q. Connect P to A and B, the intersections of circles O and Q. Then and are tangents. (OAP and OBP are right angles, since angles inscribed in semicircles are right angles.)

CONSTRUCTION 16:   To circumscribe a circle about a triangle

Given:   ABC (Fig. 15-24)

To construct:   The circumscribed circle of

Construction:   Construct the perpendicular bisectors of two sides of the triangle. Their intersection is the center of the required circle, and the distance to any vertex is the radius. (Any point on the perpendicular bisector of a segment is equidistant from the ends of the segment.)

Fig. 15-24

Fig. 15-25

CONSTRUCTION 17:   To locate the center of a given circle

Given:   A circle (Fig. 15-25)

To construct:   The center of the given circle

Construction:   Select any three points A, B, and C on the circle. Construct the perpendicular bisectors of line segments and . The intersection of these perpendicular bisectors is the center of the circle.

CONSTRUCTION 18:   To inscribe a circle in a given triangle

Given:   ABC (Fig. 15-26)

To construct:   The circle inscribed in

Construction:   Construct the bisectors of two of the angles of . Their intersection is the center of the required circle, and the distance (perpendicular) to any side is the radius. (Any point on the bisector of an angle is equidistant from the sides of the angle.)

Fig. 15-26

SOLVED PROBLEMS

15.8   Constructing tangents

A secant from a point P outside circle O in Fig. 15-27 meets the circle in B and A. Construct a triangle circumscribed about the circle so that two of its sides meet in P and the third side is tangent to the circle at A.

Solution

Use constructions 14 and 15: At A construct a tangent to circle O. From P construct tangents to circle O intersecting the first tangent in C and D. The required triangle is PCD.

Fig. 15-27

15.9   Constructing circles

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Construct the circumscribed and inscribed circles of isosceles triangle DEF in Fig. 15-2°.

Solution

Use constructions 16 and 18. In doing so, note that the bisector of E is also the perpendicular bisector of . Then the center of each circle is on . I, the center of the inscribed circle, is found by constructing the bisector of D or F. C, the center of the circumscribed circle, is found by constructing the perpendicular bisector of or .

Fig. 15-28

15.7   Inscribing and Circumscribing Regular Polygons

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CONSTRUCTION 19:   To inscribe a square in a given circle

Given:   Circle O (Fig. 15-29)

To construct:   A square inscribed in circle O

Construction:   Draw a diameter, and construct another diameter perpendicular to it. Join the end points of the diameters to form the required square.

Fig. 15-29

CONSTRUCTION 20:   To inscribe a regular octagon in a given circle

Given:   Circle O (Fig. 15-30)

To construct:   A regular octagon inscribed in circle O

Construction:   As in construction 19, construct perpendicular diameters. Then bisect the angles formed by these diameters, dividing the circle into eight congruent arcs. The chords of these arcs are the sides of the required regular octagon.

Fig. 15-30

CONSTRUCTION 21:   To inscribe a regular hexagon in a given circle

Given:   Circle O (Fig. 15-31)

To construct:   A regular hexagon inscribed in circle O

Construction:   Draw diameter and, using A and D as centers, construct four arcs having the same radius as circle O and intersecting the circle. Construct the required regular hexagon by joining consecutive points in which these arcs intersect the circle.

Fig. 15-31

Fig. 15-32

CONSTRUCTION 22:   To inscribe an equilateral triangle in a given circle

Given:   Circle O (Fig. 15-32)

To construct:   An equilateral triangle inscribed in circle O

Construction:   Inscribed equilateral triangles are obtained by joining alternately the six points of division obtained in construction 21.

15.8   Constructing Similar Triangles

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CONSTRUCTION 23:   To construct a triangle similar to a given triangle on a given line segment as base

Given:   ABC and line segment (Fig. 15-33)

To construct:    as base

Construction:   On construct using construction 2. Extend the other sides until they meet, at B. (If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.)

Fig. 15-33

SOLVED PROBLEM

15.10 Constructing similar triangles

Construct a triangle similar to triangle ABC in Fig. 15-34, with a base twice as long as the base of the given triangle.

Fig. 15-34

Fig. 15-35

Solution

Construct twice as long as , and then use construction 23.

Alternative method (Fig. 15-35): Extend two sides of to twice their lengths and join the endpoints.

SUPPLEMENTARY PROBLEMS

15.1.   Given line segments with lengths a and b as follows: . Construct a line segment whose length equals

(15.1)

15.2.   Given line segments with lengths a, b, and c: . Construct a line segment whose length equals

(15.1)

15.3.   Given angles with measures A and B (Fig. 15-36). Construct an angle with measure

(15.2)

Fig. 15-36

Fig. 15-37

15.4.   Given angles with measures A, B, and C (Fig. 15-37). Construct an angle with measure

(15.2)

15.5.   In a right triangle, construct (a) the bisector of the right angle; (b) the perpendicular bisector of the hypotenuse; (c) the median to the hypotenuse.

(15.3)

15.6.   For each kind of triangle (acute, right, and obtuse), show that the following sets of rays and segments are concurrent, that is, they intersect in one point: (a) the angle bisectors; (b) the medians; (c) the altitudes; (d) the perpendicular bisectors.

(15.3)

15.7.   Given in Fig. 15-38, construct (a) the supplement of A; (b) the complement of B; (c) the complement of C.

(15.4)

Fig. 15.38

15.8.   Construct an angle with measure equal to

(15.4)

15.9.   Given an acute angle, construct (a) its supplement; (b) its complement; (c) half its supplement; (d) half its complement.

(15.4)

15.10.   By actual construction, illustrate that the difference between the measures of the supplement and complement of an acute angle equals 90°.

(15.4)

15.11.   Construct a right triangle given its (a) legs; (b) hypotenuse and a leg; (c) leg and an acute angle adjacent to the leg; (d) leg and an acute angle opposite the leg; (e) hypotenuse and an acute angle.

(15.5)

15.12.   Construct an isosceles triangle given (a) an arm and a vertex angle; (b) an arm and a base angle; (c) an arm and the altitude to the base; (d) the base and the altitude to the base.

(15.5)

15.13.   Construct an isosceles right triangle given (a) a leg; (b) the hypotenuse; (c) the altitude to the hypotenuse.

(15.5)

15.14.   Construct a triangle given (a) two sides and the median to one of them; (b) two sides and the altitude to one of them; (c) an angle, the angle bisector of the given angle, and a side adjacent to the given angle.

(15.5)

15.15.   Construct angles of measure 158 and 165°.

(15.6)

15.16.   Given an angle with measure A, construct angles with measure (a) A + 608; (b) A + 308; (c) A + 120°.

(15.6)

15.17.   Construct a parallelogram, given (a) two adjacent sides and an angle; (b) the diagonals and the acute angle at their intersection; (c) the diagonals and a side; (d) two adjacent sides and the altitude to one of them; (e) a side, an angle, and the altitude to the given side.

(15.7)

15.18.   Circumscribe a triangle about a given circle, if the points of tangency are given.

(15.8)

15.19.   Secant passes through the center of circle O in Fig. 15-39. Circumscribe a quadrilateral about the circle so that A and B are opposite vertices.

(15.8)

Fig. 15-39

15.20.   Circumscribe and inscribe circles about (a) an acute triangle; (b) an obtuse triangle.

(15.9)

15.21.   Circumscribe a circle about (a) a right triangle; (b) a rectangle; (c) a square.

(15.9)

15.22.   Construct the inscribed and circumscribed circles of an equilateral triangle.

(15.9)

15.23.   Locate the center of a circle drawn around the outside of a half-dollar piece.

(15.9)

15.24.   In a given circle, inscribe (a) a square; (b) a regular octagon; (c) a regular 16-gon; (d) a regular hexagon; (e) an equilateral triangle; (f) a regular dodecagon.

15.25.   Construct a triangle similar to a given triangle with a base (a) three times as long; (b) half as long; (c) one and one-half times as long.

(15.10)