August 2016 Geometry Regents

1. In the diagram below, lines $\ell$ , $m$ , $n$ , and $p$ intersect line $r$ .

Which statement is true?

(1) $\ell \| n$
(2) $\ell \| p$
(3) $m \| p$
(4) $m \| n$

2. Which transformation would not always produce an image that would be congruent to the original figure?

(1) translation
(2) dilation
(3) rotation
(4) reflection

3. If an equilateral triangle is continuously rotated around one of its medians, which $3$ -dimensional object is generated?

(1) cone
(2) pyramid
(3) prism
(4) sphere

4. In the diagram below, $m\angle BDC = 100^{\circ}, m\angle A = 50^{\circ}$ , and $m\angle DBC = 30^{\circ}$ .

Which statement is true?

(1) $\triangle ABD$ is obtuse
(2) $\triangle ABC$ is isosceles
(3) $m\angle ABD = 80^{\circ}$
(4) $\triangle ABD$ is scalene

5. Which point shown in the graph below is the image of point $P$ after a counterclockwise rotation of $90^{\circ}$ about the origin?

(1) $A$
(2) $B$
(3) $C$
(4) $D$

6. In $\triangle ABC$ , where $\angle C$ is a right angle, $cos A = \dfrac{\sqrt{21}}{5}$ . What is $sin B$ ?

(1) $\dfrac{\sqrt{21}}{5}$
(2) $\dfrac{\sqrt{21}}{2}$
(3) $\dfrac{2}{5}$
(4) $\dfrac{5}{\sqrt{21}}$

7. Quadrilateral $ABCD$ with diagonals $\overline{AC}$ and $\overline{BD}$ is shown in the diagram below.

Which information is not enough to prove $ABCD$ is a parallelogram?

(1) $\overline{AB} \cong \overline{CD}$ and $\overline{AB} \| \overline{DC}$
(2) $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$
(3) $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \| \overline{AD}$
(4) $\overline{AB} \| \overline{DC}$ and $\overline{BC} \| \overline{AD}$

8. An equilateral triangle has sides of length $20$ . To the nearest tenth, what is the height of the equilateral triangle?

(1) $10.0$
(2) $11.5$
(3) $17.3$
(4) $23.1$

9. Given: $\triangle AEC, \triangle DEF$ , and $\overline{FE} \perp \overline{CE}$

What is a correct sequence of similarity transformations that shows $\triangle AEC \sim \triangle DEF$ ?

(1) a rotation of $180$ degrees about point $E$ followed by a horizontal translation
(2) a counterclockwise rotation of $90$ degrees about point $E$ followed by a horizontal translation
(3) a rotation of $180$ degrees about point $E$ followed by a dilation with a scale factor of $2$ centered at point $E$
(4) a counterclockwise rotation of $90$ degrees about point $E$ followed by a dilation with a scale factor of $2$ centered at point $E$

10. In the diagram of right triangle $ABC$ , $\overline{CD}$ intersects hypotenuse $\overline{AB}$ at $D$ .

If $AD = 4$ and $DB = 6$ , which length of $\overline{AC}$ makes $\overline{CD} \perp \overline{AB}$ ?

(1) $2\sqrt{6}$
(2) $2\sqrt{10}$
(3) $2\sqrt{15}$
(4) $4\sqrt{2}$

11. Segment $CD$ is the perpendicular bisector of $\overline{AB}$ at $E$ . Which pair of
segments does not have to be congruent?

(1) $\overline{AD}, \overline{BD}$
(2) $\overline{AC}, \overline{BC}$
(3) $\overline{AE}, \overline{BE}$
(4) $\overline{DE}, \overline{CE}$

12. In triangle $CHR, O$ is on $\overline{HR}$ , and $D$ is on $\overline{CR}$ so that $\angle H \cong \angle RDO$ .

If $RD = 4, RO = 6$ , and $OH = 4$ , what is the length of $\overline{CD}$ ?

(1) $2 \dfrac{2}{3}$
(2) $6 \dfrac{2}{3}$
(3) $11$
(4) $15$

13. The cross section of a regular pyramid contains the altitude of the pyramid. The shape of this cross section is a

(1) circle
(2) square
(3) triangle
(4) rectangle

14. The diagonals of rhombus $TEAM$ intersect at $P(2,1)$ . If the equation of the line that contains diagonal $\overline{TA}$ is $y = -x + 3$ , what is the equation of a line that contains diagonal $\overline{EM}$ ?

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

15. The coordinates of vertices $A$ and $B$ of $\triangle ABC$ are $A(3,4)$ and $B(3,12)$ . If the area of $\triangle ABC$ is $24$ square units, what could be the coordinates of point $C$ ?

(1) ( $3,6$ )
(2) ( $8,-3$ )
(3) ( $-3,8$ )
(4) ( $6,3$ )

16. What are the coordinates of the center and the length of the radius of the circle represented by the equation $x^{2} + y^{2} - 4x + 8y + 11 = 0$ ?

(1) center ( $2,-4$ ) and radius $3$
(2) center ( $-2,4$ ) and radius $3$
(3) center ( $2,-4$ ) and radius $9$
(4) center ( $-2,4$ ) and radius $9$

17. The density of the American white oak tree is $752$ kilograms per cubic meter. If the trunk of an American white oak tree has a circumference of $4.5$ meters and the height of the trunk is $8$ meters, what is the approximate number of kilograms of the trunk?

(1) $13$
(2) $9694$
(3) $13,536$
(4) $30,456$

18. Point $P$ is on the directed line segment from point $X(-6,-2)$ to point $Y(6,7)$ and divides the segment in the ratio $1:5$ . What are the coordinates of point $P$ ?

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

19. In circle $O$ , diameter $\overline{AB}$ , chord $\overline{BC}$ , and radius $\overline{OC}$ are drawn, and the measure of arc $BC$ is $108^{\circ}$ .

Some students wrote these formulas to find the area of sector $COB$ :

Amy $\dfrac{3}{10} \bullet \pi \bullet (BC)^{2}$
Beth $\dfrac{108}{360} \bullet \pi \bullet (OC)^{2}$
Carl $\dfrac{3}{10} \bullet \pi \bullet (\dfrac{1}{2}AB)^{2}$
Dex $\dfrac{108}{360} \bullet \pi \bullet \dfrac{1}{2}(AB)^{2}$

Which students wrote correct formulas?

(1) Amy and Dex
(2) Beth and Carl
(3) Carl and Amy
(4) Dex and Beth

20. Tennis balls are sold in cylindrical cans with the balls stacked one on top of the other. A tennis ball has a diameter of $6.7$ cm. To the nearest cubic centimeter, what is the minimum volume of the can that holds a stack of $4$ tennis balls?

(1) $236$
(2) $282$
(3) $564$
(4) $945$

21. Line segment $A'B'$ , whose endpoints are ( $4,-2$ ) and ( $16,14$ ), is the image of $\overline{AB}$ after a dilation of $\dfrac{1}{2}$ centered at the origin. What
is the length of $\overline{AB}$ ?

(1) $5$
(2) $10$
(3) $20$
(4) $40$

22. Given: $\triangle ABE$ and $\triangle CBD$ shown in the diagram below with $\overline{DB} \cong \overline{BE}$

Which statement is needed to prove $\triangle ABE \cong \triangle CBD$ using $SAS \cong SAS$ ?

(1) $\angle CDB \cong \angle AEB$
(2) $\angle AFD \cong \angle EFC$
(3) $\overline{AD} \cong \overline{CE}$
(4) $\overline{AE} \cong \overline{CD}$

23. In the diagram below, $\overline{BC}$ is the diameter of circle $A$ .

Point $D$ , which is unique from points $B$ and $C$ , is plotted on circle $A$ . Which statement must always be true?

(1) $\triangle BCD$ is a right triangle.
(2) $\triangle BCD$ is an isosceles triangle.
(3) $\triangle BAD$ and $\triangle CBD$ are similar triangles.
(4) $\triangle BAD$ and $\triangle CAD$ are congruent triangles.