Practice Test 2 - Geometry

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

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1. A linear transformation is represented by the matrix $Z=\begin{pmatrix}0&0\\0&0\end{pmatrix}.$ A point $V(4,-1)$ is transformed to $ZV$. Which statement describes the geometric effect?
2. Triangle $\triangle WXY$ is shown in the plane (not necessarily right). Side $\overline{WX}$ is labeled $a$ and side $\overline{WY}$ is labeled $b$. The included angle at $W$ is labeled $C$ (so $C=\angle XWY$). A dashed altitude from $Y$ meets $\overline{WX}$ at $Z$, with $\overline{YZ}\perp\overline{WX}$. Which expression represents the area of $\triangle WXY$?
3. The incircle of triangle $MNP$ has radius 4 and center $I$. If the triangle's semiperimeter is 15, what is the area of triangle $MNP$?
4. A linear transformation $T$ has matrix $M=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$ Which conclusion follows from the determinant value?
5. A drone camera measures an angle of depression $\theta$ from the horizontal to a target on level ground. The drone is 120 m above the ground, and the horizontal distance from the point directly below the drone to the target is 160 m. The model is $\tan(\theta)=\frac{120}{160}.$ Which angle solves the model within the given interval $0^\circ<\theta<90^\circ$?
6. The average monthly temperature in a city varies approximately sinusoidally over a year. The average temperature is 35°F in January (the minimum) and 85°F in July (the maximum), and the pattern repeats every 12 months. What are the amplitude, midline, and period of a sinusoidal model for the temperature (in °F) as a function of time (in months)?
7. A uniform scaling by factor $3$ about the origin is applied to point $Q(-2,5)$. Using a $2\times 2$ matrix, what is the image of $Q$?​
8. A right triangle with legs of length 6 and 8 is rotated $360°$ about its longer leg. What is the shape and approximate volume of the three-dimensional object generated?
9. In the right triangle $\triangle DEF$ shown, $\angle E$ is a right angle and the hypotenuse is $\overline{DF}$. The acute angle at $D$ is $35^\circ$, and $DE=9$. What is the length of $\overline{DF}$?
10. A linear transformation is represented by $D=\begin{pmatrix}-1&0\\0&1\end{pmatrix}.$ The unit square is shown on the coordinate plane. Which claim about area scaling is correct?
11. A student applies Cavalieri's principle to find the volume of a sphere by comparing it to a cylinder with two cones removed. The student claims that since the cross-sections match at every height, the volume formula $V = \frac{4}{3}\pi r^3$ follows immediately. What additional step is actually required?
12. A linear transformation is represented by the matrix $A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.$ A unit square has vertices $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$. Which claim about area scaling is correct?
13. In the coordinate plane, segment PQ has endpoints P(2, 4) and Q(8, 12). After a dilation centered at the origin, the image segment P'Q' has length 15 units. What was the scale factor of the dilation?
14. Two circles are drawn: $\odot E$ and $\odot F$. The center points $E$ and $F$ are marked, and the radii are different. Which statement explains why the circles are similar by referencing a dilation correctly?
15. In triangle $PQR$, the three medians intersect at point $G$. If median $PS$ has length $15$, where $S$ is the midpoint of $QR$, what is the distance from $G$ to vertex $P$?
16. In the diagram, a regular triangle $XYZ$ is inscribed in circle $\odot O$, and the center is marked. Which statement must be true about the central angles at $O$?
17. A surveyor needs to find the area of a triangular plot of land. She measures two adjacent sides as $45$ meters and $60$ meters, with an included angle of $120°$. Using an auxiliary line from the vertex of the $120°$ angle perpendicular to the opposite side, what is the area of the triangular plot?
18. In a coordinate plane, line $\ell_1$ passes through points $(1, 4)$ and $(5, 8)$, while line $\ell_2$ passes through points $(2, 3)$ and $(6, 7)$. A dilation with center $(0, 0)$ and scale factor $2$ is applied to both lines. Which statement about the images $\ell_1'$ and $\ell_2'$ is correct?
19. A right triangle $\triangle ABC$ is shown in the plane with a right angle at $A$. The hypotenuse is $BC=18$, and $\angle B = 62^\circ$. What is the length of leg $AC$?
20. In triangle $ABC$, point $D$ lies on side $\overline{BC}$, and $\overline{AD}$ is the perpendicular bisector of side $\overline{BC}$. If $AB = 13$ and $BD = 5$, what is the length of side $\overline{AC}$?

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