August 2017 Geometry Regents

1. A two-dimensional cross section is taken of a three-dimensional object. If this cross section is a triangle, what can not be the three dimensional object?

(1) cone
(2) cylinder
(3) pyramid
(4) rectangular prism

2. The image of $\triangle DEF$ is $\triangle D'E'F'$ . Under which transformation will the triangles not be congruent?

(1) a reflection through the origin
(2) a reflection over the line $y = x$
(3) a dilation with a scale factor of $1$ centered at ( $2,3$ )
(4) a dilation with a scale factor of $\dfrac{3}{2}$ centered at the origin

3. The vertices of square $RSTV$ have coordinates $R(-1,5), S(-3,1), T(-7,3),$ and $V(-5,7)$ . What is the perimeter of $RSTV$ ?

(1) $\sqrt{20}$
(2) $\sqrt{40}$
(3) $4\sqrt{20}$
(4) $4\sqrt{40}$

4. In the diagram below of circle $O$ , chord $\overline{CD}$ is parallel to diameter $\overline{AOB}$ and $m\widehat{CD} = 130$

What is $m\widehat{AC}$ ?

(1) $25$
(2) $50$
(3) $65$
(4) $115$

5. In the diagram below, $\overline{AD}$ intersects $\overline{BE}$ at $C$ , and $\overline{AB} \| \overline{DE}$ .

If $CD = 6.6$ cm, $DE = 3.4$ cm, $CE = 4.2$ cm, and $BC = 5.25$ cm, what is the length of $\overline{AC}$ , to the nearest hundredth of a centimeter?

(1) $2.70$
(2) $3.34$
(3) $5.28$
(4) $8.25$

6. As shown in the graph below, the quadrilateral is a rectangle.

Which transformation would not map the rectangle onto itself?

(1) a reflection over the $x$ -axis
(2) a reflection over the line $x = 4$
(3) a rotation of $180^{\circ}$ about the origin
(4) a rotation of $180^{\circ}$ about the point ( $4,0$ )

7. In the diagram below, triangle $ACD$ has points $B$ and $E$ on sides $\overline{AC}$ and $\overline{AD}$ , respectively, such that $\overline{BE} \| \overline{CD}, AB = 1, BC = 3.5,$ and $AD = 18$ .

What is the length of $\overline{AE}$ , to the nearest tenth?

(1) $14.0$
(2) $5.1$
(3) $3.3$
(4) $4.0$

8. In the diagram below of parallelogram $ROCK, m\angle C$ is $70^{\circ}$ and $m\angle ROS$ is $65^{\circ}$ .

What is $m\angle KSO$ ?

(1) $45^{\circ}$
(2) $110^{\circ}$
(3) $115^{\circ}$
(4) $135^{\circ}$

9. In the diagram below, $\angle GRS \cong \angle ART, GR = 36, SR = 45, AR = 15,$ and $RT = 18$ .

Which triangle similarity statement is correct?

(1) $\triangle GRS \sim \triangle ART$ by AA
(2) $\triangle GRS \sim \triangle ART$ by SAS
(3) $\triangle GRS \sim \triangle ART$ by SS
(4) $\triangle GRS$ is not similar to $\triangle ART$

10. The line represented by the equation $4y = 3x + 7$ is transformed by a dilation centered at the origin. Which linear equation could represent its image?

(1) $14.0$
(2) $5.1$
(3) $3.3$
(4) $4.0$

11. Given $\triangle ABC$ with $m\angle B = 62^{\circ}$ and side $\overline{AC}$ extended to $D$ as shown below.

Which value of $x$ makes $\overline{AB} \cong \overline{CB}$ ?

(1) $59^{\circ}$
(2) $62^{\circ}$
(3) $118^{\circ}$
(4) $121^{\circ}$

12. In the diagram shown below, $\overline{PA}$ is tangent to circle $T$ at $A$ , and secant $\overline{PBC}$ is drawn where point $B$ is on circle $T$ .

If $PB = 3$ and $BC = 15$ , what is the length of $\overline{PA}$ ?

(1) $3\sqrt{5}$
(2) $3\sqrt{6}$
(3) $3$
(4) $9$

13. A rectangle whose length and width are $10$ and $6$ , respectively, is shown below. The rectangle is continuously rotated around a straight line to form an object whose volume is $150\pi$ .

Which line could the rectangle be rotated around?

(1) a long side
(2) a short side
(3) the vertical line of symmetry
(4) the horizontal line of symmetry

14. If $ABCD$ is a parallelogram, which statement would prove that $ABCD$ is a rhombus?

(1) $\angle ABC \cong \angle CDA$
(2) $\overline{AC} \cong \overline{BD}$
(3) $\overline{AC} \perp \overline{BD}$
(4) $\overline{AB} \perp \overline{CD}$

15. To build a handicapped-access ramp, the building code states that for every $1$ inch of vertical rise in height, the ramp must extend out $12$ inches horizontally, as shown in the diagram below.

What is the angle of inclination, $x$ , of this ramp, to the nearest hundredth of a degree?

(1) $4.76$
(2) $4.78$
(3) $85.22$
(4) $85.24$

16. In the diagram below of $\triangle ABC, D, E,$ and $F$ are the midpoints of $\overline{AB}, \overline{BC},$ and $\overline{CA}$ , respectively.

What is the ratio of the area of $\triangle CFE$ to the area of $\triangle CAB$ ?

(1) $1:1$
(2) $1:2$
(3) $1:3$
(4) $1:4$

17. The coordinates of the endpoints of $\overline{AB}$ are $A(-8, -2)$ and $B(16, 6)$ . Point $P$ is on $\overline{AB}$ . What are the coordinates of point $P$ , such that $AP:PB$ is $3:5$ ?

(1) ( $1, 1$ )
(2) ( $7, 3$ )
(3) ( $9.6, 3.6$ )
(4) ( $6.4, 2.8$ )

18. Kirstie is testing values that would make triangle $KLM$ a right triangle when $\overline{LN}$ is an altitude, and $KM = 16$ as shown below.

Which lengths would make triangle KLM a right triangle?

(1) $LM = 13$ and $KN = 6$
(2) $LM = 12$ and $KN = 9$
(3) $KL = 11$ and $KN = 7$
(4) $LN = 8$ and $NM = 10$

19. In right triangle $ABC, m\angle A = 32^{\circ}, m\angle B = 90^{\circ},$ and $AC = 6.2$ cm. What is the length of $\overline{BC}$ , to the nearest tenth of a centimeter?

(1) $3.3$
(2) $3.9$
(3) $5.3$
(4) $11.7$

20. The 2010 U.S. Census populations and population densities are shown in the table below.

Based on the table above, which list has the states’ areas, in square miles, in order from largest to smallest?

(1) Illinois, Florida, New York, Pennsylvania
(2) New York, Florida, Illinois, Pennsylvania
(3) New York, Florida, Pennsylvania, Illinois
(4) Pennsylvania, New York, Florida, Illinois

21. In a right triangle, $sin (40 - x)^{\circ} = cos (3x)^{\circ}$ . What is the value of $x$ ?

(1) $10$
(2) $15$
(3) $20$
(4) $25$

22. A regular decagon is rotated $n$ degrees about its center, carrying the decagon onto itself. The value of $n$ could be

(1) $10^{\circ}$
(2) $150^{\circ}$
(3) $225^{\circ}$
(4) $252^{\circ}$

23. In a circle with a diameter of $32$ , the area of a sector is $\dfrac{512\pi}{3}$ . The measure of the angle of the sector, in radians, is

(1) $\dfrac{\pi}{3}$
(2) $\dfrac{4\pi}{3}$
(3) $\dfrac{16\pi}{3}$
(4) $\dfrac{64\pi}{3}$

24. What is an equation of the perpendicular bisector of the line segment shown in the diagram below?

(1) $y + 2x = 0$
(2) $y - 2x = 0$
(3) $2y + x = 0$
(4) $2y - x = 0$