# Comprehensive Benchmark Assessment Series

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2 CO-HS.SDG.1a.i State precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. (CCSS: G-CO.1) 1) What is defined below? : a portion of a line bounded by two points arc axis ray segment CO-HS.SDG.1a.iii Describe transformations as functions that take points in the plane as inputs and give other points as outputs. (CCSS: G-CO.2 2) A transformation takes point A to point B. Which transformation(s) could it be? F only F and R only F and T only F, R, and T 2

3 CO-HS.SDG.1a.v Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (CCSS: G-CO.3) 3) Which transformation will place the trapezoid onto itself? counterclockwise rotation about the origin by 90 rotation about the origin by 180 reflection across the x-axis reflection across the y-axis 3

4 CO-HS.SDG.1a.viii Specify a sequence of transformations that will carry a given figure onto another. (CCSS: G-CO.5) 4) Which transformation maps the solid figure onto the dashed figure? rotation 180 about the origin translation to the right and down reflection across the x-axis reflection across the y-axis 4

5 CO-HS.SDG.1b.i Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. (CCSS: G-CO.6) 5) Figure 1 is reflected about the x-axis and then translated four units left. Which figure results? Figure A Figure B Figure C Figure D 5

6 CO-HS.SDG.1b.ii Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. (CCSS: G-CO.6) 6) It is known that a series of rotations, translations, and reflections superimposes sides a, b, and c of Quadrilateral X onto three sides of Quadrilateral Y. Which is true about z, the length of the fourth side of Quadrilateral Y? 6

7 CO-HS.SDG.1b.iii Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. (CCSS: G-CO.7) 7) The triangle below can be subject to reflections, rotations, or translations. With which of the triangles can it coincide after a series of these transformations? Figures are not necessarily drawn to scale. 7

8 CO-HS.SDG.1b.iv Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (CCSS: G-CO.8) 8) Given the information regarding triangles ABC and DEF, which statement is true? The given information matches the SAS criterion; the triangles are congruent. The given information matches the ASA criterion; the triangles are congruent. Angles C and F are also congruent; this must be shown before using the ASA criterion. It cannot be shown that the triangles are necessarily congruent. 8

9 CO-HS.SDG.1c.i Prove theorems about lines and angles. (CCSS: G-CO.9) 9) Which statements should be used to prove that the measures of angles 1 and 5 sum to 180? Angles 1 and 8 are congruent as corresponding angles; angles 5 and 8 form a linear pair. Angles 1 and 2 form a linear pair; angles 3 and 4 form a linear pair. Angles 5 and 7 are congruent as vertical angles; angles 6 and 8 are congruent as vertical angles. Angles 1 and 3 are congruent as vertical angles; angles 7 and 8 form a linear pair. CO-HS.SDG.1c.ii Prove theorems about triangles. (CCSS: G-CO.10) 10)

10 CO-HS.SDG.1c.iii Prove theorems about parallelograms. (CCSS: G-CO.11) 11) Two pairs of parallel line form a parallelogram. Becki proved that angles 2 and 6 are congruent. She is first used corresponding angles created by a transversal and then alternate interior angles. Which pairs of angles could she use? 1 and 2 then 5 and 6 4 and 2 then 4 and 6 7 and 2 then 7 and 6 8 and 2 then 8 and 6 CO-HS.SDG.1c.iii Prove theorems about parallelograms. (CCSS: G-CO.11) 12) To prove that diagonals of a parallelogram bisect each other, Xavier first wants to establish that triangles APD and CPB are congruent. Which criterion and elements can he use? SAS: sides AP & PD and CP & PB with the angles in between SAS: sides AD & AP and CB & CP with the angles in between ASA: sides DP and PB with adjacent angles ASA: sides AD and BC with adjacent angles 10

11 CO-HS.SDG.1d.i Make formal geometric constructions with a variety of tools and methods. (CCSS: G-CO.12) 13) Which triangle was constructed congruent to the given triangle? Triangle 1 Triangle 2 Triangle 3 Triangle 4 11

12 CO-HS.SDG.1d.i Make formal geometric constructions with a variety of tools and methods. (CCSS: G-CO.12) 14) What is the first step in constructing congruent angles? Draw ray DF. From point A, draw an arc that intersects the sides of the angle at point B and C. From point D, draw an arc that intersects the sides of the angle at point E and F. From points A and D, draw equal arcs that intersects the rays AC and DF. CO-HS.SDG.1d.ii Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (CCSS: G-CO.13) 15) The radius of circle O is r. A circle with the same radius drawn around P intersects circle O at point R. What is the measure of angle ROP?

13 CO-HS.SDG.1d.ii Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (CCSS: G-CO.13) 16) Carol is constructing an equilateral triangle with P and R being two of the vertices. She is going to use a compass to draw circles around P and R. What should the radius of the circles be? CO-HS.SDG.2a.i.1 Show that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. (CCSS: G-SRT.1a) 17) Line b is defined by the equation y = 8 - x. If line b undergoes a dilation with a scale factor of 0.5 and center P, which equation will define the image of the line? y = 4 - x y = 5 - x y = 8 - x y = 11 - x 13

14 CO-HS.SDG.2a.i.2 Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. (CCSS: G-SRT.1b) 18) GH = 1. A dilation with center H and a scale factor of 0.5 is applied. What will be the length of the image of the segment GH? CO-HS.SDG.2a.ii Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. (CCSS: G-SRT.2) 19) When are two triangles considered similar but not congruent? The distance between corresponding vertices are equal. The distance between corresponding vertices are proportionate. The vertices are reflected across the x-axis. Each of the vertices are shifted up by the same amount. 14

15 CO-HS.SDG.2a.iii Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. (CCSS: G-SRT.2) 20) Triangle ABC was reflected and dilated so that it coincides with triangle XYZ. How did this transformation affect the sides and angles of triangle ABC? 15

16 CO-HS.SDG.2b.i Prove theorems about triangles. (CCSS: G-SRT.4) 21) Lines AC and FG are parallel. Which statement should be used to prove that triangles ABC and DBE are similar? Angles BDE and BCA are congruent as alternate interior angles. Angles BAC and BEF are congruent as corresponding angles. Angles BED and BCA are congruent as corresponding angles. Angles BDG and BEF are congruent as alternate exterior angles. CO-HS.SDG.2b.ii Prove that all circles are similar. (CCSS: G-C.1) 22) Which can be accomplished using a sequence of similarity transformations? I. mapping circle O onto circle P so that O 1 matches P 1 II. mapping circle P onto circle O so that P 1 matches O 1 I only II only both I and II neither I nor II 16

17 CO-HS.SDG.2b.iii Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (CCSS: G-SRT.5) 23) Given the diagram below, what is the value of x?

18 CO-HS.SDG.2b.iii Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (CCSS: G-SRT.5) 24) A scale model of the Millennium Dome in Greenwich, England, was constructed on a scale of 100 meters to 1 foot. The cable supports are 50 meters high and form a triangle with the cables. How high are the cable supports on the scale model that was built? 0.5 foot 1 foot 1.5 feet 2 feet 18

19 CO-HS.SDG.2c.i Explain that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. (CCSS: G-SRT.6) 25) What is the sine ratio of / P in the given triangle? CO-HS.SDG.2c.ii Explain and use the relationship between the sine and cosine of complementary angles. (CCSS: G-SRT.7) 26) Which is equal to sin 30? cos 30 cos 60 sin 60 sin 70 19

20 CO-HS.SDG.2c.iii Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. * (CCSS: G-SRT.8) 27) A new sidewalk was built to expand the flowerbed, as shown in the diagram below. About how many extra meters is the new sidewalk? 3 meters 5 meters 6 meters 8 meters CO-HS.SDG.2c.iii Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. * (CCSS: G-SRT.8) 28) A 30-foot long escalator forms a 41 angle at the second floor. Which is the closest height of the first floor? 20 feet 22.5 feet 24.5 feet 26 feet 20

21 CO-HS.SDG.2e.i Identify and describe relationships among inscribed angles, radii, and chords. (CCSS: G-C.2) 29) CO-HS.SDG.2e.ii Construct the inscribed and circumscribed circles of a triangle. (CCSS: G-C.3) 30) The center of the inscribed circle of a triangle has been established. Which point on one of the sides of a triangle should be chosen to set the width of the compass? intersection of the side and the median to that side intersection of the side and the angle bisector of the opposite angle intersection of the side and the perpendicular passing through the center intersection of the side and the altitude dropped from the opposite vertex 21

22 CO-HS.SDG.2e.iii Prove properties of angles for a quadrilateral inscribed in a circle. (CCSS: G-C.3) 31) CO-HS.SDG.2e.iii Prove properties of angles for a quadrilateral inscribed in a circle. (CCSS: G-C.3) 32) Quadrilateral ABCD is inscribed in a circle. Segments AB and BC are not the same length. Segment AC is a diameter. Which must be true? ABCD is a trapezoid. ABCD is a rectangle. ABCD has at least two right angles. ABCD has an axis of symmetry. 22

23 CO-HS.SDG.2f.ii Derive the formula for the area of a sector. (CCSS: G-C.5) 33) What is the area of the shaded sector? 23

24 CO-HS.SDG.2f.ii Derive the formula for the area of a sector. (CCSS: G-C.5) 34) What is the area of the 90 sector? 24

25 CO-HS.SDG.2f.ii Derive the formula for the area of a sector. (CCSS: G-C.5) 35) What is the area of the shaded sector if the radius of circle Z is 5 inches? 25

26 CO-HS.SDG.2f.ii Derive the formula for the area of a sector. (CCSS: G-C.5) 36) What is the area of the shaded sector, given circle Q has a diameter of 10? 26

27 CO-HS.SDG.3a.ii.1 Use coordinates to prove simple geometric theorems algebraically. (CCSS: G-GPE.4) 37) 27

28 CO-HS.SDG.3a.ii.2 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (CCSS: G-GPE.5) 38) Which statement is true about the two lines whose equations are given below? The lines are perpendicular. The lines are parallel. The lines coincide. The lines intersect, but are not perpendicular. CO-HS.SDG.3a.ii.3 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (CCSS: G-GPE.6) 39) Given A(0, 0) and B(60, 60), what are the coordinates of point M that lies on segment AB, such that AM:MB = 2:3? (24, 24) (24, 36) (40, 40) (40, 90) CO-HS.SDG.3a.ii.4 Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles. * (CCSS: G-GPE.7) 40) Two of the vertices of a triangle are (0, 1) and (4, 1). Which coordinates of the third vertex make the area of the triangle equal to 16? (0, -9) (0, 5) (4, -7) (4, -3) 28

29 CO-HS.SDG.4a.i Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. (CCSS: G-GMD.1) 41) To estimate the area of a circle, Irene divided the circle into 30 congruent sectors. Then she combined pairs of sectors into shapes as shown below. As the shapes resemble rectangles, she treats the shapes as rectangles with the height r (radius) and the base equal to the length of the curved side of one sector. What is the area of each shape? 29

30 CO-HS.SDG.4a.i Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. (CCSS: G-GMD.1) 42) The prism can be cut into three pyramids with the shaded faces congruent. If the shaded faces are considered as bases, then all three pyramids have the same height, h. Therefore the pyramids have equal volumes. What is the volume of each pyramid? 30

31 CO-HS.SDG.4a.ii Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.? (CCSS: G-GMD.3) 43) Find the volume of the cylinder cubic cm cubic cm cubic cm 75.4 cubic cm CO-HS.SDG.4a.ii Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.? (CCSS: G-GMD.3) 44) Find the volume of the rectangular pyramid. 72 cubic inches 200 cubic inches 320 cubic inches 960 cubic inches 31

32 CO-HS.SDG.5a.i Use geometric shapes, their measures, and their properties to describe objects.? (CCSS: G-MG.1) 45) The diameter of one side of a 10-foot log is approximately 13 inches. The diameter of the other side of the log is approximately 11 inches. Which is the best way to estimate the volume (in cubic feet) of the log? CO-HS.SDG.5a.ii Apply concepts of density based on area and volume in modeling situations.? (CCSS: G-MG.2) 46) Given the size and mass of each of the solid cubes X and Y, how many times is the density of cube X greater than the density of cube Y?

33 CO-HS.SDG.5a.iii Apply geometric methods to solve design problems.? (CCSS: G-MG.3) 47) Stephanie is going to form a clay model of the moon. The model will have a diameter of 2 feet, and the clay she will use comes in containers as described below. What is the least number of containers Stephanie will need in order to complete the model?

34 CO-HS.SDG.5a.iii Apply geometric methods to solve design problems.? (CCSS: G-MG.3) 48) Lewis is going to form a clay model of a skyscraper. The model will be in the shape of a 2-foot tall prism with a 6-inch by 6-inch base. The clay he will use comes in containers as described below. What is the least number of containers Lewis will need in order to complete the model? Copyright 2012 Assessment Technology, Incorporated. All rights reserved. 34

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