June 2017 Geometry Regents

1. In the diagram below, $\triangle ABC \cong \triangle DEF$ .

Which sequence of transformations maps $\triangle ABC$ onto $\triangle DEF$ ?

(1) a reflection over the $x$ -axis followed by a translation
(2) a reflection over the $y$ -axis followed by a translation
(3) a rotation of $180^{\circ}$ about the origin followed by a translation
(4) a counterclockwise rotation of $90^{\circ}$ about the origin followed by
a translation

2. On the set of axes below, the vertices of $\triangle PQR$ have coordinates $P(6,7), Q(2,1)$ , and $R(1,3)$ .

What is the area of $\triangle PQR$ ?

(1) $10$
(2) $20$
(3) $25$
(4) $50$

3. In right triangle $ABC, m\angle C = 90^{\circ}$ . If $cos B = \dfrac{5}{13}$ , which function also equals $\dfrac{5}{13}$ ?

(1) $tan A$
(2) $tan B$
(3) $sin A$
(4) $sin B$

4. In the diagram below, $m \widehat{ABC} = 268^{\circ}$ .

What is the number of degrees in the measure of $\angle ABC$ ?

(1) $134^{\circ}$
(2) $92^{\circ}$
(3) $68^{\circ}$
(4) $46^{\circ}$

5. Given $\triangle MRO$ shown below, with trapezoid $PTRO, MR = 9, MP = 2,$ and $PO = 4$ .

What is the length of $\overline{TR}$ ?

(1) $4.5$
(2) $5$
(3) $3$
(4) $6$

6. A line segment is dilated by a scale factor of $2$ centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true?

(1) The line segments are perpendicular, and the image is one-half of the length of the given line segment.
(2) The line segments are perpendicular, and the image is twice the length of the given line segment.
(3) The line segments are parallel, and the image is twice the length of the given line segment.
(4) The line segments are parallel, and the image is one-half of the length of the given line segment.

7. Which figure always has exactly four lines of reflection that map the figure onto itself?

(1) square
(2) rectangle
(3) regular octagon
(4) equilateral triangle

8. In the diagram below of circle $O$ , chord $\overline{DF}$ bisects chord $\overline{BC}$ at $E$ .

If $BC = 12$ and $FE$ is $5$ more than $DE$ , then $FE$ is

(1) $13$
(2) $9$
(3) $6$
(4) $4$

9. Kelly is completing a proof based on the figure below.

She was given that $\angle A \cong \angle EDF$ , and has already proven $\overline{AB} \cong \overline{DE}$ . Which pair of corresponding parts and triangle congruency method would not prove $\triangle ABC \cong \triangle DEF$ ?

(1) $\overline{AC} \cong \overline{DF}$ and SAS
(2) $\overline{BC} \cong \overline{EF}$ and SAS
(3) $\angle C \cong \angle F$ and AAS
(4) $\angle CBA \cong \angle FED$ and ASA

10. In the diagram below, $\overline{DE}$ divides $\overline{AB}$ and $\overline{AC}$ proportionally, $m\angle C = 26^{\circ}, m\angle A = 82^{\circ},$ and $\overline{DF}$ bisects $\angle BDE$ .

The measure of angle $DFB$ is

(1) $36^{\circ}$
(2) $54^{\circ}$
(3) $72^{\circ}$
(4) $82^{\circ}$

11. Which set of statements would describe a parallelogram that can always be classified as a rhombus?

I. Diagonals are perpendicular bisectors of each other.
II. Diagonals bisect the angles from which they are drawn.
III. Diagonals form four congruent isosceles right triangles.

(1) I and II
(2) I and III
(3) II and III
(4) I, II, and III

12. The equation of a circle is $x^{2} + y^{2} - 12y + 20 = 0$ . What are the
coordinates of the center and the length of the radius of the circle?

(1) center ( $0,6$ ) and radius $4$
(2) center ( $0,-6$ ) and radius $4$
(3) center ( $0,6$ ) and radius $16$
(4) center ( $0,-6$ ) and radius $16$

13. In the diagram of $\triangle RST$ below, $m\angle T = 90^{\circ}, RS = 65,$ and $ST = 60$ .

What is the measure of $\angle S$ , to the nearest degree?

(1) $23^{\circ}$
(2) $43^{\circ}$
(3) $47^{\circ}$
(4) $67^{\circ}$

14. Triangle $A'B'C'$ is the image of $\triangle ABC$ after a dilation followed by a translation.
Which statement(s) would always be true with respect to this sequence of transformations?

I. $\triangle ABC \cong \triangle A'B'C'$
II. $\triangle ABC \sim \triangle A'B'C'$
III. $\overline{AB} \| \overline{A'B'}$
IV. $AA' = BB'$

(1) II, only
(2) I and II
(3) II and III
(4) II, III, and IV

15. Line segment $RW$ has endpoints $R(-4,5)$ and $W(6,20)$ . Point $P$ is on $\overline{RW}$ such that $RP:PW$ is $2:3$ . What are the coordinates of point $P$ ?

(1) ( $2,9$ )
(2) ( $0,11$ )
(3) ( $2,14$ )
(4) ( $10,2$ )

16. The pyramid shown below has a square base, a height of $7$ , and a volume of $84$ .

What is the length of the side of the base?

(1) $6$
(2) $12$
(3) $18$
(4) $36$

17. In the diagram below of triangle $MNO, \angle M$ and $\angle O$ are bisected by $\overline{MS}$ and $\overline{OR}$ , respectively. Segments $MS$ and $OR$ intersect at $T$ , and $m\angle N = 40^{\circ}$ .

If $m\angle TMR = 28^{\circ}$ , the measure of angle $OTS$ is

(1) $40^{\circ}$
(2) $50^{\circ}$
(3) $60^{\circ}$
(4) $70^{\circ}$

18. In the diagram below, right triangle $ABC$ has legs whose lengths are
$4$ and $6$ .

What is the volume of the three-dimensional object formed by continuously rotating the right triangle around $\overline{AB}$ ?

(1) $32\pi$
(2) $48\pi$
(3) $96\pi$
(4) $144\pi$

19. What is an equation of a line that is perpendicular to the line whose equation is $2y = 3x - 10$ and passes through ( $-6,1$ )?

(1) $y = -\dfrac{2}{3}x - 5$
(2) $y = -\dfrac{2}{3}x - 3$
(3) $y = \dfrac{2}{3}x + 1$
(4) $y = \dfrac{2}{3}x + 10$

20. In quadrilateral $BLUE$ shown below, $\overline{BE} \cong \overline{UL}$ .

Which information would be sufficient to prove quadrilateral $BLUE$ is a parallelogram?

(1) $\overline{BL} \| \overline{EU}$
(2) $\overline{LU} \| \overline{BE}$
(3) $\overline{BE} \cong \overline{BL}$
(4) $\overline{LU} \cong \overline{EU}$

21. A ladder $20$ feet long leans against a building, forming an angle of $71^{\circ}$ with the level ground.
To the nearest foot, how high up the wall of the building does the ladder touch the building?

(1) $15$
(2) $16$
(3) $18$
(4) $19$

22. In the two distinct acute triangles $ABC$ and $DEF$ , $\angle B \cong \angle E$ . Triangles $ABC$ and $DEF$ are congruent when there is a sequence of rigid motions that maps

(1) $\angle A$ onto $\angle D$ , and $\angle C$ onto $\angle F$
(2) $\overline{AC}$ onto $\overline{DF}$ , and $\overline{BC}$ onto $\overline{EF}$
(3) $\angle C$ onto $\angle F$ , and $\overline{BC}$ onto $\overline{EF}$
(4) point $A$ onto point $D$ , and $\overline{AB}$ onto $\overline{DE}$

23. A fabricator is hired to make a $27$ -foot-long solid metal railing for the stairs at the local library. The railing is modeled by the diagram below. The railing is $2.5$ inches high and $2.5$ inches wide and is comprised of a rectangular prism and a half-cylinder.