Practice Test 12 - Geometry

20 questions. Answer them in any order, then submit to see your score and every explanation.

Start practice test

1. A student is constructing a line parallel to $\ell$ through a point $P$ using only a compass and straightedge. The student copies an angle formed by transversal $\overline{AP}$ with line $\ell$ to create a congruent angle at $P$. Which reasoning explains why the construction works?
2. A spherical balloon has radius $0.5\text{ m}$. Which expression represents the volume of the balloon in cubic meters?
3. A transformation is defined by $A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}.$ What does this transformation do to a point $(x, y)$?
4. Square $PQRS$ is rotated $45°$ counterclockwise about its center to create square $P'Q'R'S'$. Then square $P'Q'R'S'$ is reflected across a line passing through the center of the original square to create square $P''Q''R''S''$. If the final image $P''Q''R''S''$ has the same orientation as the original square $PQRS$, what was the direction of the reflection line?
5. A rectangle RSTU is shown with vertices $R(-2,-3)$, $S(4,-3)$, $T(4,1)$, and $U(-2,1)$. Which calculation is required to find the area?
6. A storage tank is modeled as a rectangular prism with interior dimensions $4\text{ m} \times 3\text{ m} \times 2\text{ m}$. A gas fills the tank with an energy density of $18\ \text{kJ/m}^3$. Which expression represents the total energy stored in the gas?
7. Two circles intersect at exactly two points. A student concludes that because the circles intersect, they cannot be similar since similar figures must be non-intersecting. Which statement best describes this reasoning?
8. Points $N(-1,0)$ and $O(7,8)$ are plotted on a coordinate plane and connected by segment $\overline{NO}$. Point $V$ divides the directed segment from $N$ to $O$ internally in the ratio $NV:VO=5:3$. Which coordinates represent the partition point $V$?
9. A sphere of radius $R$ is compared to a solid formed by taking a cylinder of radius $R$ and height $2R$ and removing two congruent cones (each cone has base radius $R$ and height $R$). Slices are taken by horizontal planes at equal heights $y$ from the center. Which claim is NOT supported by the slicing argument?
10. A non-right triangle $\triangle XYZ$ has $XY=6$, $XZ=11$, and included angle $\angle YXZ=30^\circ$ emphasized at $X$. Which setup correctly uses a justification aligned with the Law of Cosines (and its Pythagorean special case) to find $YZ$?
11. Triangle $\triangle PQR$ has vertices $P(1,1)$, $Q(3,1)$, and $R(2,3)$. A transformation is given by $T=\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}.$ What are the coordinates of the image triangle $\triangle P'Q'R'$?
12. A right triangle $\triangle JKL$ is shown in the plane. $\angle K$ is marked as a right angle. The hypotenuse is $\overline{JL}$ and is labeled $25$. The leg $\overline{KL}$ is labeled $7$. The diagram is not drawn to scale. No other angles or lengths are marked. What is the length of $\overline{JK}$?
13. Triangle $PQR$ is reflected across line $\ell$ to produce triangle $P'Q'R'$. Then triangle $P'Q'R'$ is reflected across line $m$, where lines $\ell$ and $m$ intersect at point $O$ and form a $35°$ angle. The final image is triangle $P''Q''R''$. What single rigid motion is equivalent to this composition of two reflections?
14. In the plane, right triangle $\triangle GHI$ is shown. The right angle is explicitly marked at $H$ ($\angle GHI = 90^\circ$). The hypotenuse is $\overline{GI}$ and is labeled $15$. The leg $\overline{GH}$ is labeled $9$. The leg $\overline{HI}$ is unlabeled. Which method should be used to find the length of $\overline{HI}$? (Diagram is not drawn to scale. No acute angle measures are given.)
15. A student constructs the perpendicular bisector of $\overline{AB}$ by drawing equal-radius circles centered at $A$ and $B$ that intersect at $P$ and $Q$, then drawing line $\overleftrightarrow{PQ}$. Which statement correctly describes the constructed line $\overleftrightarrow{PQ}$?
16. Consider triangles $PQR$ and $STU$ where it is known that $\angle P = 42°$, $\angle Q = 73°$, $\angle S = 42°$, and $\angle T = 65°$. A student concludes that since both triangles have an angle of $42°$, and the AA criterion only requires two pairs of congruent angles, she just needs to find one more pair. Her reasoning about establishing similarity is:
17. A parabola has vertex at $(-1, 3)$ and passes through the point $(3, 7)$. If the parabola opens upward, what is the equation of its directrix?
18. Refer to the figure below. In right triangle $PQR$ with the right angle at $Q$, $PQ = 5$ and $QR = 12$. What is $\cos(R)$?
19. On the coordinate plane, the focus is $F(5,1)$ and the directrix is the vertical line $x=1$ (shown). The parabola opens to the right. Which statement identifies the vertex?
20. A community center is installing a rectangular stage on a rectangular floor area that is $24\text{ ft}$ wide and $18\text{ ft}$ deep. Building code requires a clear walkway of at least $3\text{ ft}$ on all four sides of the stage. The stage must be a rectangle with side lengths that are whole numbers of feet. Which design satisfies all constraints?

Practice tests