Posttest

This test is your chance to bring together all the material you have learned through this book. It may ask you to combine ideas in new ways. In the multiple-choice section, choose the best answer for each question. In the free response questions, show all your work, and explain your thinking as clearly as possible.

PART 1

Questions 1 to 30 are multiple-choice questions. Choose the best answer for each question.

1.   What transformation would NOT map Triangle I onto Triangle II?

A)  Reflection across the line y = x

B)  180° rotation about the point (½, ½)

C)  Translation 5 units right and 5 units down

D)  Reflection across y =½, followed by reflection across x = ½

2.   The diagram shows circle O with secants and drawn from point P outside the circle. Chord is also shown. m∠QRT = 47°, m = 74°, and m∠VUT = 103°. What is the measure of ∠RPV?

A)  20°

B)  30°

C)  40°

D)  60°

3.   In ΔABC, is parallel to and divides and in ratio 1:3. EC = 21 cm, DB = 12 cm, and BC = 24 cm. Find the perimeter of ΔADE to the nearest centimeter.

A)  17

B)  19

C)  22

D)  35

4.   PQRT is a trapezoid with and . m∠Q = 108°. If ΔSRT is isosceles with vertex angle ∠T, what is the measure of ∠SRT?

A)  72°

B)  108°

C)  144°

D)  36°

5.   Which of the following polyhedrons, if sliced parallel to its base, could NOT have a square cross section?

A)  Cube

B)  Rectangular prism

C)  Triangular prism

D)  Square pyramid

6.   In right triangle ΔABC, with , sin(∠C) = . Which of these is also equal to ?

A)  sin (∠A)

B)  sin(∠B)

C)  cos(∠A)

D)  cos(∠C)

7.   ΔABC is not drawn to scale. Side is extended through point C to point D. m∠BCD = 115°. Which of the following is a true statement?

A)  m∠ABC = 115°

B)  m∠ACB = m∠BAC

C)  m∠ABC + m∠BCA =115°

D)  m∠ABC + m∠BAC =115°

8.   In right triangle ΔABC with right angle ∠B, is drawn perpendicular to hypotenuse . If AD = 4 cm, and DC = 12 cm, find the perimeter of ΔABC to the nearest tenth of a centimeter.

A)  13.9

B)  21.9

C)  37.9

D)  55.4

9.   The vertices of ΔRST have coordinates R(6,3), S(4,1), and T(0,7). What is the perimeter of ΔRST, to the nearest tenth?

A)  10.0

B)  17.2

C)  13.0

D)  14.5

10.   If quadrilateral MNOP is a parallelogram, which of these is not sufficient to prove MNOP is a rectangle?

A)  MO = NP

B)  

C)  

D)  ∠M ≅ ∠N

11.   Solve for x: sin(2x + 7) = cos(4x − 1)

A)  4

B)  11.5

C)  14

D)  19

12.   In the diagram of circle O, and intersect at point E, m∠AEC = 42° and = 51°. What is ?

A)  9°

B)  33°

C)  42°

D)  51°

13.   ΔABC is a right triangle with right angle ∠B. . If AD, DB, AF, and FC have the measures shown and EC = , how long is ?

A)  3

B)  

C)  7

D)  

14.   The Point Arena Light in northern California is a 115-foot concrete tower, built on a rocky point 40 feet above sea level, placing the light 155 feet above sea level. The light can be seen more than 25 miles out to sea. If a ship is 1 mile out to sea, what is the measurement of the angle of elevation to the top of the lighthouse, to the nearest degree? (1 mile = 5,280 feet)

A)  1°

B)  2°

C)  42°

D)  89

15.   The coordinates of the endpoints of are X(1, 9) and Y(4, −1). Find the coordinates of point P that divides into two segments whose lengths are in ratio XP:PY = 2:3.

A)  (1, 5)

B)  (2.2, 7.4)

C)  (2.2, 5)

D)  (2.2, 0.6)

16.   The image of ΔRST under a reflection is ΔR’S’T’. The pre-image and image are shown. Which of the following equations represents the line of reflection?

A)  y = x

B)  y = − x

C)  y = 9 − x

D)  y = x − 9

17.   Line is parallel to . Transversals and intersect at point E as shown. Which of these conclusions can be drawn?

A)  ΔABE ≅ ΔCED

B)  ΔABE ≅ ΔDEC

C)  ΔABE ~ ΔCED

D)  ΔABE ~ ΔDCE

Use the diagram below for questions 18 and 19.

18.   In ΔRST, RS = 21 cm, RT = 16 cm, and m∠R = 80°. What is the area of ΔRST to the nearest tenth of a square centimeter?

A)  165.4

B)  168.0

C)  29.2

D)  952.8

19.   Using the figure in question 18, find the height SV of ΔRST to the nearest tenth of a centimeter.

A)  10.3

B)  12.0

C)  18.3

D)  20.7

20.   A regular octagon is rotated about its center. Which of these rotations will NOT map the octagon onto itself?

A)  45° clockwise

B)  60° counterclockwise

C)  90° clockwise

D)  180° counterclockwise

21.   A right triangle with legs of 18 cm and 24 cm is rotated about a line to create a cone with a volume of 2,592π cubic centimeters. About which of these lines is the triangle rotated to produce a cone with that volume?

A)  The 18 cm leg

B)  The 24 cm leg

C)  The hypotenuse

D)  The altitude to the hypotenuse

22.   Line segment l is shown in the diagram. Which of these points does not lie on the perpendicular bisector of line segment l?

A)  (4, 0)

B)  (1, 4)

C)  (7, −4)

D)  (3, 1)

23.   The table above shows four states with similar population density, in people per square mile, and their land area in square miles. Rank the states according to population, from smallest to largest.

A)  Indiana, Georgia, New Hampshire, Louisiana

B)  Louisiana, New Hampshire, Georgia, Indiana

C)  Georgia, Indiana, Louisiana, New Hampshire

D)  New Hampshire, Louisiana, Indiana, Georgia

24.   Two triangles are shown below. Markings on the diagram show that and ∠C ≅ ∠Y. Which of the following statements must be shown to be true in order to prove ΔABC ≅ ΔXZY ?

A)  ∠A ≅ ∠X

B)  

C)  

D)  ∠A ≅ ∠Z

25.   Rhombus PQRS has diagonals and that intersect at point T. PR = 24 cm and QS = 9 cm. Find the perimeter of PQRS to the nearest tenth of a centimeter.

A)  12.8

B)  31.2

C)  51.2

D)  60.0

26.   Which of these is the equation of a circle of radius 7 centered at the point (3, −4)?

A)  

B)  

C)  

D)  

27.   In ΔABC, m∠A = 45°and m∠B = 110°. In ΔRST, m∠R = 45° and m∠ S = 25°. Which statement about these two triangles is correct?

A)  ΔABC ~ ΔRST

B)  ΔABC ~ ΔRTS

C)  ΔABC ~ ΔTRS

D)  ΔABC and ΔRST have no similarity relationship.

28.   Circle O has a diameter of 30 cm. ∠AOB is a central angle, and the sector defined by ∠AOB has an area of 20π. What is the length of arc to the nearest tenth of a centimeter?

A)  2.1

B)  8.4

C)  32.0

D)  62.8

Use the figure below for questions 29 and 30.

29.   The diagram above shows part of the work of constructing the perpendicular bisector of segment . What is the next necessary step in the construction?

A)  Draw

B)  Draw

C)  Draw an arc centered at C, passing through D

D)  Bisect

30.   Which of these statements provides the rationale for the construction of the perpendicular bisector of segment ?

A)  Two distinct points determine a unique line.

B)  Every perpendicular line bisects the segment to which it is drawn.

C)  Every point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

D)  A line bisects a segment if and only if it is perpendicular to the segment.

PART 2

Answer all questions in this part. Show your work and explain your reasoning.

31.   The points A(1,1), B(3,2), C(3,5), and D(1, 4) are the vertices of a quadrilateral. The quadrilateral is transformed by reflection across the x-axis, followed by a dilation by a factor of 2 centered at the origin. Draw the original quadrilateral and its image under these transformations.

32.   Use compass and straightedge to construct a regular hexagon inscribed in the circle below.

33.   ABCD is an isosceles trapezoid and F is the midpoint of . ∠EAD ≅ ∠EDA. Prove that ΔAFE ≅ ΔDFE.

34.   Parallelogram ABCD is drawn in the coordinate plane with the vertices shown. Use the coordinates given to prove or disprove the claim: The line segment connecting the midpoints of opposite sides of parallelogram ABCD bisects both diagonals.

35.   The diagram shows a cone and two pyramids, one with a regular hexagon as its base, and one with a square base. All three figures have the same height and the same radius. (The radius of a pyramid is the distance from the center of the regular polygon to a vertex.) Which of these figures has the greatest volume? Explain your reasoning.

36.   Find the center and radius of the circle whose equation is . Sketch its graph.

37.   The Johnson family’s home is built on a trapezoidal lot as shown in the diagram. The house is a rectangle, and a trapezoidal deck was added to one side of the house. A paved driveway in the shape of a parallelogram leads from the edge of the property to a small rectangular garage. There is a square garden shed in one corner of the property. The Johnsons want to add a pool to their property. They would like the pool to be as large as possible, but it cannot come in contact with any existing structure or the edge of the property. Give the Johnsons your best recommendation for the shape, size, and location of the pool. Use the coordinate system to locate and measure. Explain the reasoning behind your recommendation.

38.   The Pentagon, the building that houses the headquarters of the U.S. Department of Defense, is a regular pentagon 281 meters on a side. The central courtyard, also a regular pentagon, has an area of 20,000 square meters. What is the area of the building itself to the nearest square meter? How long is the side of the courtyard in meters?