January 2016 Geometry Regents

1. William is drawing pictures of cross sections of the right circular cone below.

Which drawing can not be a cross section of a cone?

2. An equation of a line perpendicular to the line represented by the equation $y = -\dfrac{1}{2}x - 5$ and passing through ( $6, -4$ ) is

(1) $y = -\dfrac{1}{2}x + 4$
(2) $y = -\dfrac{1}{2}x - 1$
(3) $y = 2x + 14$
(4) $y = 2x - 16$

3. In parallelogram $QRST$ shown below, diagonal $\overline{TR}$ is drawn, $U$ and $V$ are points on $\overline{TS}$ and $\overline{QR}$ , respectively, and $\overline{UV}$ intersects $\overline{TR}$ at $W$ .

If $m\angle S = 60^{\circ}, m\angle SRT = 83^{\circ}$ , and $m\angle TWU = 35^{\circ}$ , what is $m\angle WVQ$ ?

(1) $37^{\circ}$
(2) $60^{\circ}$
(3) $72^{\circ}$
(4) $83^{\circ}$

4. A fish tank in the shape of a rectangular prism has dimensions of $14$ inches, $16$ inches, and $10$ inches. The tank contains $1680$ cubic inches of water. What percent of the fish tank is empty?

(1) $10$
(2) $25$
(3) $50$
(4) $75$

5. Which transformation would result in the perimeter of a triangle being different from the perimeter of its image?

(1) $(x, y) \rightarrow (y,x)$
(2) $(x, y) \rightarrow (x, -y)$
(3) $(x, y) \rightarrow (4x, 4y)$
(4) $(x, y) \rightarrow (x + 2, y - 5)$

6. In the diagram below, $\overleftrightarrow{FE}$ bisects $\overline{AC}$ at $B$ , and $\overleftrightarrow{GE}$ bisects $\overline{BD}$ at $C$ .

Which statement is always true?

(1) $\overline{AB} \cong \overline{DC}$
(2) $\overline{FB} \cong \overline{EB}$
(3) $\overleftrightarrow{BD}$ bisects $\overline{GE}$ at $C$
(4) $\overleftrightarrow{AC}$ bisects $\overline{FE}$ at $B$

7. As shown in the diagram below, a regular pyramid has a square base whose side measures $6$ inches.

If the altitude of the pyramid measures $12$ inches, its volume, in cubic inches, is

(1) $72$
(2) $144$
(3) $288$
(4) $432$

8. Triangle $ABC$ and triangle $DEF$ are graphed on the set of axes below.

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

(1) a reflection over the $x$ -axis followed by a reflection over the $y$ -axis
(2) a $180^{\circ}$ rotation about the origin followed by a reflection over the line $y = x$
(3) a $90^{\circ}$ clockwise rotation about the origin followed by a reflection over the $y$ -axis
(4) a translation $8$ units to the right and $1$ unit up followed by a $90^{\circ}$ counterclockwise rotation about the origin

9. In $\triangle ABC$ , the complement of $\angle B$ is $\angle A$ . Which statement is always true?

(1) $tan \angle A = tan \angle B$
(2) $sin \angle A = sin \angle B$
(3) $cos \angle A = tan \angle B$
(4) $sin \angle A = cos \angle B$

10. A line that passes through the points whose coordinates are ( $1,1$ ) and ( $5,7$ ) is dilated by a scale factor of $3$ and centered at the origin. The image of the line

(1) is perpendicular to the original line
(2) is parallel to the original line
(3) passes through the origin
(4) is the original line

11. Quadrilateral $ABCD$ is graphed on the set of axes below.

When $ABCD$ is rotated $90^{\circ}$ in a counterclockwise direction about the origin, its image is quadrilateral $A'B'C'D'$ . Is distance preserved
under this rotation, and which coordinates are correct for the given vertex?

(1) no and $C'(1,2)$
(2) no and $D'(2,4)$
(3) yes and $A'(6,2)$
(4) yes and $B'(-3,4)$

12. In the diagram below of circle $O$ , the area of the shaded sector $LOM$ is $2\pi cm^{2}$ .

If the length of $\overline{NL}$ is $6$ cm, what is $m\angle N$ ?

(1) $10^{\circ}$
(2) $20^{\circ}$
(3) $40^{\circ}$
(4) $80^{\circ}$

13. In the diagram below, $\triangle ABC \sim \triangle DEF$ .

If $AB = 6$ and $AC = 8$ , which statement will justify similarity by SAS?

(1) $DE = 9, DF = 12,$ and $\angle A \cong \angle D$
(2) $DE = 8, DF = 10,$ and $\angle A \cong \angle D$
(3) $DE = 36, DF = 64,$ and $\angle C \cong \angle F$
(4) $DE = 15, DF = 20,$ and $\angle C \cong \angle F$

14. The diameter of a basketball is approximately $9.5$ inches and the diameter of a tennis ball is approximately $2.5$ inches. The volume of the basketball is about how many times greater than the volume of the tennis ball?

(1) $3591$
(2) $65$
(3) $55$
(4) $4$

15. The endpoints of one side of a regular pentagon are ( $1,4$ ) and ( $2,3$ ). What is the perimeter of the pentagon?

(1) $\sqrt{10}$
(2) $5\sqrt{10}$
(3) $5\sqrt{2}$
(4) $25\sqrt{2}$

16. In the diagram of right triangle $ABC$ shown below, $AB = 14$ and $AC = 9$ .

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

(1) $33$
(2) $40$
(3) $50$
(4) $57$

17. What are the coordinates of the center and length of the radius of the circle whose equation is $x^{2} + 6x + y^{2} - 4y = 23$ ?

(1) ( $3,-2$ ) and $36$
(2) ( $3,-2$ ) and $6$
(3) ( $-3,2$ ) and $36$
(4) ( $-3,2$ ) and $6$

18. The coordinates of the vertices of $\triangle RST$ are $R(-2, -3), S(8,2)$ , and $T(4,5)$ . Which type of triangle is $\triangle RST$ ?

(1) right
(2) acute
(3) obtuse
(4) equiangular

19. Molly wishes to make a lawn ornament in the form of a solid sphere. The clay being used to make the sphere weighs $.075$ pound per cubic inch. If the sphere’s radius is $4$ inches, what is the weight of the sphere, to the nearest pound?

(1) $34$
(2) $20$
(3) $15$
(4) $4$

20. The ratio of similarity of $\triangle BOY$ to $\triangle GRL$ is $1:2$ . If $BO = x + 3$ and $GR = 3x - 1$ , then the length of $\overline{GR}$ is

(1) $5$
(2) $7$ (3) $10$
(4) $20$

21. In the diagram below, $\overline{DC}$ , $\overline{AC}$ , $\overline{DOB}$ , $\overline{CB}$ , and $\overline{AB}$ are chords of circle $O$ , $\overleftrightarrow{FDE}$ is tangent at point $D$ , and radius $\overline{AO}$ is drawn. Sam decides to apply this theorem to the diagram: “An angle inscribed in a semi-circle is a right angle.”

Which angle is Sam referring to?

(1) $\angle AOB$
(2) $\angle BAC$
(3) $\angle DCB$
(4) $\angle FDB$

22. In the diagram below, $\overline{CD}$ is the altitude drawn to the hypotenuse $\overline{AB}$ of right triangle $ABC$ .

Which lengths would not produce an altitude that measures $6\sqrt{2}$ ?

(1) $AD = 2$ and $DB = 36$
(2) $AD = 3$ and $AB = 24$
(3) $AD = 6$ and $DB = 12$
(4) $AD = 8$ and $AB = 17$

23. A designer needs to create perfectly circular necklaces. The necklaces each need to have a radius of $10$ cm. What is the largest number of necklaces that can be made from $1000$ cm of wire?

(1) $15$
(2) $16$
(3) $31$
(4) $32$

24. In $\triangle SCU$ shown below, points $T$ and $O$ are on $\overline{SU}$ and $\overline{CU}$ , respectively. Segment $OT$ is drawn so that $\angle C \cong \angle OTU$ .

If $TU = 4, OU = 5$ , and $OC = 7$ , what is the length of $\overline{ST}$ ?

(1) $5.6$
(2) $8.75$
(3) $11$
(4) $15$