January 2018 Geometry Regents

1. In the diagram below, a sequence of rigid motions maps $ABCD$ onto $JKLM$ .

If $m\angle A = 82^{\circ}, m\angle B = 10^{\circ}$ , and $m\angle L = 121^{\circ}$ , the measure of $\angle M$ is

(1) $53^{\circ}$
(2) $82^{\circ}$
(3) $104^{\circ}$
(4) $121^{\circ}$

2. Parallelogram $HAND$ is drawn below with diagonals $\overline{HN}$ and $\overline{AD}$ intersecting at $S$ .

Which statement is always true?

(1) $HN = \dfrac{1}{2}AD$
(2) $AS = \dfrac{1}{2}AD$
(3) $\angle AHS \cong \angle ANS$
(4) $\angle HDS \cong \angle NDS$

3. The graph below shows two congruent triangles, $ABC$ and $A'B'C'$ .

Which rigid motion would map $\triangle ABC$ onto $\triangle A'B'C'$ ?

(1) a rotation of $90$ degrees counterclockwise about the origin
(2) a translation of three units to the left and three units up
(3) a rotation of $180$ degrees about the origin
(4) a reflection over the line $y = x$

4. A man was parasailing above a lake at an angle of elevation of $32^{\circ}$ from a boat, as modeled in the diagram below.

If $129.5$ meters of cable connected the boat to the parasail, approximately how many meters above the lake was the man?

(1) $68.6$
(2) $80.9$
(3) $109.8$
(4) $244.4$

5. A right hexagonal prism is shown below. A two-dimensional cross section that is perpendicular to the base is taken from the prism.

Which figure describes the two-dimensional cross section?

(1) triangle
(2) rectangle
(3) pentagon
(4) hexagon

6. In the diagram below, $\overline{AC}$ has endpoints with coordinates $A(-5,2)$ and $C(4,-10)$ .

If $B$ is a point on $\overline{AC}$ and $AB:BC = 1:2$ , what are the coordinates of $B$ ?

(1) ( $-2, -2$ )
(2) ( $-\dfrac{1}{2}, -4$ )
(3) ( $0, -\dfrac{14}{3}$ )
(4) ( $1, -6$ )

7. An ice cream waffle cone can be modeled by a right circular cone with a base diameter of $6.6$ centimeters and a volume of $54.45\pi$ cubic centimeters. What is the number of centimeters in the height of the waffle cone?

(1) $3\dfrac{3}{4}$
(2) $5$
(3) $15$
(4) $24\dfrac{3}{4}$

8. The vertices of $\triangle PQR$ have coordinates $P(2,3), Q(3,8)$ , and $R(7,3)$ . Under which transformation of $\triangle PQR$ are distance and angle measure preserved?

(1) $(x,y) \rightarrow (2x,3y)$
(2) $(x,y) \rightarrow (x + 2, 3y)$
(3) $(x,y) \rightarrow (2x, y + 3)$
(4) $(x,y) \rightarrow (x + 2, y + 3)$

9. In $\triangle ABC$ shown below, side $\overline{AC}$ is extended to point $D$ with $m\angle DAB = (180 - 3x)^{\circ}, m\angle B = (6x - 40)^{\circ},$ and $m\angle C = (x + 20)^{\circ}$ .

What is $m/angle BAC$ ?

(1) $20^{\circ}$
(2) $40^{\circ}$
(3) $60^{\circ}$
(4) $80^{\circ}$

10. Circle $O$ is centered at the origin. In the diagram below, a quarter of circle $O$ is graphed.

Which three-dimensional figure is generated when the quarter circle is continuously rotated about the $y$ -axis?

(1) cone
(2) sphere
(3) cylinder
(4) hemisphere

11. Rectangle $A'B'C'D'$ is the image of rectangle $ABCD$ after a dilation centered at point A by a scale factor of $\dfrac{2}{3}$ . Which statement is correct?

(1) Rectangle $A'B'C'D'$ has a perimeter that is $\dfrac{2}{3}$ the perimeter of rectangle $ABCD$
(2) Rectangle $A'B'C'D'$ has a perimeter that is $\dfrac{3}{2}$ the perimeter of rectangle $ABCD$
(3) Rectangle $A'B'C'D'$ has an area that is $\dfrac{2}{3}$ the area of rectangle $ABCD$
(4) Rectangle $A'B'C'D'$ has an area that is $\dfrac{3}{2}$ the area of rectangle $ABCD$

12. The equation of a circle is $x^{2} + y^{2} - 6x + 2y = 6$ . What are the coordinates of the center and the length of the radius of the circle?

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

13. In the diagram of $\triangle ABC$ below, $\overline{DE}$ is parallel to $\overline{AB}, CD = 15, AD = 9,$ and $AB = 40$ .

The length of $\overline{DE}$ is

(1) $15$
(2) $24$
(3) $25$
(4) $30$

14. The line whose equation is $3x - 5y = 4$ is dilated by a scale factor of $\dfrac{5}{3}$ centered at the origin. Which statement is correct?

(1) The image of the line has the same slope as the pre-image but a different $y$ -intercept.
(2) The image of the line has the same $y$ -intercept as the pre-image but a different slope.
(3) The image of the line has the same slope and the same $y$ -intercept as the pre-image.
(4) The image of the line has a different slope and a different $y$ -intercept from the pre-image.

15. Which transformation would not carry a square onto itself?

(1) a reflection over one of its diagonals
(2) a $90^{\circ}$ rotation clockwise about its center
(3) a $180^{\circ}$ rotation about one of its vertices
(4) a reflection over the perpendicular bisector of one side

16. In circle $M$ below, diameter $\overline{AC}$ , chords $\overline{AB}$ and $\overline{BC}$ , and radius $\overline{MB}$ are drawn.

Which statement is not true?

(1) $\triangle ABC$ is a right triangle
(2) $\triangle ABM$ is isosceles
(3) $m\widehat{BC} = m\angle BMC$
(4) $m\widehat{AB} = \dfrac{1}{2}m\angle ACB$

17. In the diagram below, $\overline{XS}$ and $\overline{YR}$ intersect at $Z$ . Segments $XY$ and $RS$ are drawn perpendicular to $\overline{YR}$ to form triangles $XYZ$ and $SRZ$ .

Which statement is always true?

(1) $(XY)(SR) = (XZ)(RZ)$
(2) $\triangle XYZ \cong \triangle SRZ$
(3) $\overline{XS} \cong \overline{YR}$
(4) $\dfrac{XY}{SR} = \dfrac{YZ}{RZ}$

18. As shown in the diagram below, $\overline{ABC} \| \overline{EFG}$ and $\overline{BF} \cong \overline{EF}$ .

If $m\angle CBF = 42.5^{\circ}$ , then $m\angle EBF$ is

(1) $42.5^{\circ}$
(2) $68.75^{\circ}$
(3) $95^{\circ}$
(4) $137.5^{\circ}$

19. A parallelogram must be a rhombus if its diagonals

(1) are congruent
(2) bisect each other
(3) do not bisect its angles
(4) are perpendicular to each other

20. What is an equation of a line which passes through ( $6,9$ ) and is perpendicular to the line whose equation is $4x - 6y = 15$ ?

(1) $y - 9 = -\dfrac{3}{2}(x - 6)$
(2) $y - 9 = \dfrac{2}{3}(x - 6)$
(3) $y + 9 = -\dfrac{3}{2}(x + 6)$
(4) $y + 9 = \dfrac{2}{3}(x + 6)$

21. Quadrilateral $ABCD$ is inscribed in circle $O$ , as shown below.

If $m\angle A = 80^{\circ}, m\angle B = 75^{\circ}, m\angle C = (y + 30)^{\circ},$ and $m\angle D = (x - 10)^{\circ}$ , which statement is true?

(1) $x = 85$ and $y = 50$
(2) $x = 90$ and $y = 45$
(3) $x = 110$ and $y = 75$
(4) $x = 115$ and $y = 70$

22. A regular pyramid has a square base. The perimeter of the base is $36$ inches and the height of the pyramid is $15$ inches. What is the volume of the pyramid in cubic inches?

(1) $180$
(2) $405$
(3) $540$
(4) $1215$

23. In the diagram below of $\triangle ABC, \angle ABC$ is a right angle, $AC = 12, AD = 8$ , and altitude $\overline{BD}$ is drawn.

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

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

24. In the diagram below, two concentric circles with center $O$ , and radii $\overline{OC}, \overline{OD}, \overline{OCE},$ and $\overline{ODF}$ are drawn.

If $OC = 4$ and $OE = 6$ , which relationship between the length of arc $EF$ and the length of arc $CD$ is always true?

(1) The length of arc $EF$ is $2$ units longer than the length of arc $CD$ .
(2) The length of arc $EF$ is $4$ units longer than the length of arc $CD$ .
(3) The length of arc $EF$ is $1.5$ times the length of arc $CD$ .
(4) The length of arc $EF$ is $2.0$ times the length of arc $CD$ .