January 2019 Geometry Regents

1. After a dilation with center ( $0,0$ ), the image of $\overline{DB}$ is $\overline{D'B'}$ . If $DB = 4.5$ and $D'B' = 18$ , the scale factor of this dialtion is

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

2. In the diagram below, $\triangle ABC$ with sides of $13$ , $15$ , and $16$ , is mapped onto $\triangle DEF$ after a clockwise rotation of $90^{\circ}$ about point $P$ .

If $DE = 2x - 1$ , what is the value of $x$ ?

(1) $7$
(2) $7.5$
(3) $8$
(4) $8.5$

3. On the set of axes below, $\triangle ABC$ has vertices at $A(-2,0)$ , $B(2,-4)$ , $C(4,2)$ , and $\triangle DEF$ has vertices at $D(4,0)$ , $E(-4,8)$ , $F(-8,-4)$ .

Which sequence of transformations will map $\triangle ABC$ onto $\triangle DEF$ ?

(1) a dilation of $\triangle ABC$ by a scale factor of $2$ centered at point $A$
(2) a dilation of $\triangle ABC$ by a scale factor of $\dfrac{1}{2}$ centered at point $A$
(3) a dilation of $\triangle ABC$ by a scale factor of $2$ centered at the origin, followed by a rotation of $180^{\circ}$ about the origin
(4) a dilation of $\triangle ABC$ by a scale factor of $\dfrac{1}{2}$ centered at the origin, followed by a rotation of $180^{\circ}$ about the origin

4. The figure below shows a rhombus with noncongruent diagonals.

Which transformation would not carry this rhombus onto itself?

(1) a reflection over the shorter diagonal
(2) a reflection over the longer diagonal
(3) a clockwise rotation of $90^{\circ}$ about the intersection of the diagonals
(4) a counterclockwise rotation of $180^{\circ}$ about the intersection of the diagonals

5. In the diagram below of circle $O$ , points $K$ , $A$ , $T$ , $I$ , and $E$ are on the circle, $\triangle KAE$ and $\triangle ITE$ are drawn, $\widehat{KE} \cong \widehat{EI}$ and $\angle EKA \cong \angle EIT$ .

Which statement about $\triangle KAE$ and $\triangle ITE$ is always true?

(1) They are neither congruent nor similar.
(2) They are similar but not congruent.
(3) They are right triangles.
(4) They are congruent.

6. In right triangle $ABC$ shown below, point $D$ is on $\overline{AB}$ and point $E$ is on $\overline{CB}$ such that $\overline{AC} \| \overline{DE}$ .

If $AB = 15$ , $BC = 12$ , and $EC = 7$ , what is the length of $\overline{BD}$ ?

(1) $8.75$
(2) $6.25$
(3) $5$
(4) $4$

7. In rhombus $VENU$ , diagonals $\overline{VN}$ and $\overline{EU}$ intersect at $S$ .
If $VN = 12$ and $EU = 16$ , what is the perimeter of the rhombus?

(1) $80$
(2) $40$
(3) $20$
(4) $10$

8. Given right triangle $ABC$ with a right angle at $C$ , $m\angle B = 61^{\circ}$ . Given right triangle $RST$ with a right angle $T$ , $m\angle R = 29^{\circ}$ .

Which proportion in relation to $\triangle ABC$ and $\triangle RST$ is not correct?

(1) $\dfrac{AB}{RS} = \dfrac{RT}{AC}$
(2) $\dfrac{BC}{ST} = \dfrac{AB}{RS}$
(3) $\dfrac{BC}{ST} = \dfrac{AC}{RT}$
(4) $\dfrac{AB}{AC} = \dfrac{RS}{RT}$

9. A vendor is using an $8$ -ft by $8$ -ft tent for a craft fair. The legs of the tent are $9$ ft tall and the top forms a square pyramid with a height of $3$ ft.

What is the volume, in cubic feet, of space the tent occupies?

(1) $256$
(2) $640$
(3) $672$
(4) $768$

10. In the diagram below of right triangle $KMI$ , altitude $\overline{IG}$ is drawn to hyotenuse $\overline{KM}$ .

If $KG = 9$ and $IG = 12$ , the length of $\overline{IM}$ is

(1) $15$
(2) $16$
(3) $20$
(4) $25$

11. Which three-dimensional figure will result when a rectangle $6$ inches long and $5$ inches wide is continuously rotated about the longer side?

(1) a rectangular prism with a length of $6$ inches, width of $6$ inches, and height of $5$ inches
(2) a rectangular prism with a length of $6$ inches, width of $5$ inches, and height of $5$ inches
(3) a cylinder with a radius of $5$ inches and a height of $6$ inches
(4) a cylinder with a radius of $6$ inches and a height of $5$ inches

12. Which statement about parallelograms is always true?

(1) The diagonals are congruent.
(2) The diagonals bisect each other.
(3) The diagonals are perpendicular.
(4) The diagonals bisect their respective angles.

13. From a point on the ground one-half mile from the base of a historic monument, the angle of elevation to its top is $11.87^{\circ}$ . To the nearest foot, what is the height of the monument?

(1) $543$
(2) $555$
(3) $1086$
(4) $1110$

14. The area of a sector of a circle with a radius measuring $15$ cm is $75\pi$ cm^{2 .}What is the measure of the central angle that forms the sector?

(1) $72^{\circ}$
(2) $120^{\circ}$
(3) $144^{\circ}$
(4) $180^{\circ}$

15. Point $M$ divides $\overline{AB}$ so that $AM:MB = 1:2$ . If $A$ has coordinates ( $-1, -3$ ) and $B$ has coordinates ( $8, 9$ ), the coordinates of $M$ are

(1) ( $2, 1$ )
(2) ( $\dfrac{5}{3}, 0$ )
(3) ( $5, 5$ )
(4) ( $\dfrac{23}{3}, 8$ )

16. In the diagram below of triangle $ABC$ , $\overline{AC}$ is extended through point $C$ to point $D$ is drawn to $\overline{AC}$ .

Which equation is always true?

(1) $m\angle 1 = m\angle 3 + m\angle 2$
(2) $m\angle 5 = m\angle 3 - m\angle 2$
(3) $m\angle 6 = m\angle 3 - m\angle 2$
(4) $m\angle 7 = m\angle 3 + m\angle 2$

17. In the diagram below of right triangle $ABC$ , $AC = 8$ , and $AB = 17$ .

Which equation would determine the value of angle $A$ ?

(1) $sin A = \dfrac{8}{17}$
(2) $tan A = \dfrac{8}{15}$
(3) $cos A = \dfrac{15}{17}$
(4) $tan A = \dfrac{15}{8}$

18. Francisco needs the three pieces of glass shown below to complete a stained glass window. The shapes, two triangles, and a trapezoid, are measured in inches.

Glass can be purchased in rectangular sheets that are $12$ inches wide. What is the minimum length of a sheet of glass, in inches, that Francisco must purchase in order to have enough to complete the window?

(1) $20$
(2) $25$
(3) $29$
(4) $34$

19. In the diagram of quadrilateral $NAVY$ below, $m\angle YNA = 30^{\circ}$ , $m\angle YAN = 38^{\circ}$ , $m\angle AVY= 94^{\circ}$ , and $m\angle VAY= 46^{\circ}$ .

Which segment has the shortest length?

(1) $\overline{AY}$
(2) $\overline{NY}$
(3) $\overline{VA}$
(4) $\overline{VY}$

20. What is an equation of a circle whose center is ( $1,4$ ) and diameter is $10$ ?

(1) $x^{2} - 2x + y^{2} - 8y = 8$
(2) $x^{2} + 2x + y^{2} + 8y = 8$
(3) $x^{2} - 2x + y^{2} - 8y = 83$
(4) $x^{2} + 2x + y^{2} + 8y = 83$

21. On the set of axes below, $\triangle ABC$ , altitude $\overline{CG}$ , and median $\overline{CM}$ are drawn.

Which expression represents the area of $\triangle ABC$ ?

(1) $\dfrac{(BC)(AC)}{2}$
(2) $\dfrac{(GC)(BC)}{2}$
(3) $\dfrac{(CM)(AB)}{2}$
(4) $\dfrac{(GC)(AB)}{2}$

22. In right triangle $ABC$ , $m\angle C = 90^{\circ}$ and $AC \neq BC$ . Which trigonometric ratio is equivalent to $sin B$ ?

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

23. As shown in the diagram below, the radius of a cone is $2.5$ cm and its slant height is $6.5$ cm.

How many cubic centimeters are in the volume of the cone?

(1) $12.5\pi$
(2) $13.5\pi$
(3) $30.0\pi$
(4) $37.5\pi$

24. What is an equation of the image of the line $y = \dfrac{3}{2}x - 4$ after a dilation of a scale factor of $\dfrac{3}{4}$ centered at the origin?

(1) $y = \dfrac{9}{8}x - 4$
(2) $y = \dfrac{9}{8}x - 3$
(3) $y = \dfrac{3}{2}x - 4$
(4) $y = \dfrac{3}{2}x - 3$