1. A parallelogram must be a rectangle when its

(1) diagonals are perpendicular
(2) diagonals are congruent
(3) opposite sides are parallel
(4) opposite sides are congruent

2. If $\triangle A'B'C'$ is the image of $\triangle ABC$, under which transformation will the triangles not be congruent?

(1) reflection over the $x$-axis
(2) translation to the left $5$ and down $4$
(3) dilation centered at the origin with scale factor $2$
(4) rotation of $270^{\circ}$ counterclockwise about the origin

3. If the rectangle below is continuously rotated about side $w$, which solid figure is formed?

(1) pyramid
(2) rectangular prism
(3) cone
(4) cylinder

4. Which expression is always equivalent to $sin x$ when $0^{\circ} \textless x \textless 90^{\circ}$

(1) $cos (90^{\circ} - x)$
(2) $cos (45^{\circ} - x)$
(3) $cos (2x)$
(4) $cos x$

5. In the diagram below, a square is graphed in the coordinate plane.

A reflection over which line does not carry the square onto itself?

(1) $x = 5$
(2) $y = 2$
(3) $y = x$
(4) $x + y = 4$

6. The image of $\triangle ABC$ after a dilation of scale factor $k$ centered at point $A$ is $\triangle ADE$, as shown in the diagram below.

Which statement is always true?

(1) $2AB = AD$
(2) $\overline {AD} \perp \overline{DE}$
(3) $AC = CE$
(4) $\overline{BC} \| \overline{DE}$

7. A sequence of transformations maps rectangle $ABCD$ onto rectangle $A''B''C''D''$, as shown in the diagram below.

Which sequence of transformations maps $ABCD$ onto $A'B'C'D'$ and then maps $A'B'C'D'$ onto $A''B''C''D''$?

(1) a reflection followed by a rotation
(2) a reflection followed by a translation
(3) a translation followed by a rotation
(4) a translation followed by a reflection

8. In the diagram of parallelogram $FRED$ shown below, $\overline{ED}$ is extended to $A$, and $\overline{AF}$ is drawn such that $\overline{AF} \cong \overline{DF}$.

If $m\angle R = 124^{\circ}$, what is $m\angle AFD$?

(1) $124^{\circ}$
(2) $112^{\circ}$
(3) $68^{\circ}$
(4) $56^{\circ}$

9. If $x^{2} + 4x + y^{2} - 6y - 12 = 0$ is the equation of a circle, the length of the radius is

(1) $25$
(2) $16$
(3) $5$
(4) $4$

10. Given $\overline{MN}$ hown below, with $M (-6, 1)$ and $N (3, -5)$, what is an equation of the line that passes through point $P(6,1)$ and is parallel to $\overline{MN}$?

(1) $y = -\dfrac{2}{3}x + 5$
(2) $y = -\dfrac{2}{3}x - 3$
(3) $y = \dfrac{3}{2}x + 7$
(4) $y = \dfrac{3}{2}x - 8$

11. Linda is designing a circular piece of stained glass with a diameter of $7$ inches. She is going to sketch a square inside the circular region.

To the nearest tenth of an inch, the largest possible length of a side of the square is

(1) $3.5$
(2) $4.9$
(3) $5.0$
(4) $6.9$

12. In the diagram shown below, $\overline{AC}$ is tangent to circle $O$ at $A$ and to circle $P$ at $C$, $\overline{OP}$ intersects $\overline{AC}$ at $B, OA = 4, AB = 5,$ and $PC = 10$.

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

(1) $6.4$
(2) $8$
(3) $12.5$
(4) $16$

13. In the diagram below, which single transformation was used to map triangle $A$ onto triangle $B$?

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

14. In the diagram below, $\triangle DEF$ is the image of $\triangle ABC$ after a clockwise rotation of $180^{\circ}$ and a dilation where $AB = 3, BC = 5.5, AC = 4.5, DE = 6, FD = 9,$ and $EF = 11$.

Which relationship must always be true?

(1) $\dfrac{m\angle A}{m\angle D} = \dfrac{1}{2}$
(2) $\dfrac{m\angle C}{m\angle F} = \dfrac{2}{1}$
(3) $\dfrac{m\angle A}{m\angle C} = \dfrac{m\angle F}{m\angle D}$
(4) $\dfrac{m\angle B}{m\angle E} = \dfrac{m\angle C}{m\angle F}$

15. In the diagram below, quadrilateral $ABCD$ is inscribed in circle $P$.

What is $m\angle ADC$?

(1) $70^{\circ}$
(2) $72^{\circ}$
(3) $108^{\circ}$
(4) $110^{\circ}$

16. A hemispherical tank is filled with water and has a diameter of $10$ feet. If water weighs $62.4$ pounds per cubic foot, what is the total weight of the water in a full tank, to the nearest pound?

(1) $16,336$
(2) $32,673$
(3) $130,690$
(4) $261,381$

17. In the diagram below, $\triangle ABC \sim \triangle ADE$.

Which measurements are justified by this similarity?

(1) $AD = 3, AB = 6, AE = 4,$ and $AC = 12$
(2) $AD = 5, AB = 8, AE = 7,$ and $AC = 10$
(3) $AD = 3, AB = 9, AE = 5,$ and $AC = 10$
(4) $AD = 2, AB = 6, AE = 5,$ and $AC = 15$

18. Triangle $FGH$ is inscribed in circle $O$, the length of radius $\overline{OH}$ is $6$, and $\overline{FH} \cong \overline{OG}$.

What is the area of the sector formed by angle $FOH$?

(1) $2\pi$
(2) $\dfrac{3}{2}\pi$
(3) $6\pi$
(4) $24\pi$

19. As shown in the diagram below, $\overline{AB}$ and $\overline{CD}$ intersect at $E$, and $\overline{AC} \| \overline{BD}$.

Given $\triangle AEC \sim \triangle BED$, which equation is true?

(1) $\dfrac{CE}{DE} = \dfrac{EB}{EA}$
(2) $\dfrac{AE}{BE} = \dfrac{AC}{BD}$
(3) $\dfrac{EC}{AE} = \dfrac{BE}{ED}$
(4) $\dfrac{ED}{EC} = \dfrac{AC}{BD}$

20. A triangle is dilated by a scale factor of $3$ with the center of dilation at the origin. Which statement is true?

(1) The area of the image is nine times the area of the original triangle.
(2) The perimeter of the image is nine times the perimeter of the original triangle.
(3) The slope of any side of the image is three times the slope of the corresponding side of the original triangle.
(4) The measure of each angle in the image is three times the measure of the corresponding angle of the original triangle.

21. The Great Pyramid of Giza was constructed as a regular pyramid with a square base. It was built with an approximate volume of $2,592,276$ cubic meters and a height of $146.5$ meters. What was the length of one side of its base, to the nearest meter?

(1) $73$
(2) $77$
(3) $133$
(4) $230$

22. A quadrilateral has vertices with coordinates ($-3, 1$), ($0, 3$), ($5, 2$), and ($-1, -2$). Which type of quadrilateral is this?

(1) rhombus
(2) rectangle
(3) square
(4) trapezoid

23. In the diagram below, $\triangle ABE$ is the image of $\triangle ACD$ after a dilation centered at the origin. The coordinates of the vertices are $A(0,0), B(3,0), C(4.5,0), D(0,6),$ and $E(0,4)$.

The ratio of the lengths of $\overline{BE}$ to $\overline{CD}$ is

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

24. Line $y = 3x - 1$ is transformed by a dilation with a scale factor of $2$ and centered at ($3,8$). The line’s image is

(1) $y = 3x - 8$
(2) $y = 3x - 4$
(3) $y = 3x - 2$
(4) $y = 3x - 1$