Practice Test 4 - Geometry

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

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1. A matrix $M=\begin{pmatrix}2&0\\0&3\end{pmatrix}$ is applied to a unit square (a square of area 1). Which claim about area scaling is correct?
2. Line $\ell_1$ passes through points $(2, 6)$ and $(8, 3)$. Line $\ell_2$ is the reflection of $\ell_1$ across the x-axis. What is the slope of line $\ell_2$?
3. A wildlife reserve is modeled by the triangular region shown (see diagram). The base is labeled $10\text{ km}$ and the perpendicular height to that base is labeled $6\text{ km}$ (a right-angle box marks the height as perpendicular to the base). The animal density is $18$ animals per square kilometer. Which calculation gives the total number of animals in the reserve?
4. On the coordinate plane, points $A(-4,1)$ and $B(2,7)$ are connected by segment $\overline{AB}$. Point $P$ lies on the directed segment from $A$ to $B$ and divides $\overline{AB}$ internally in the ratio $AP:PB=1:2$. Which point divides the segment in the given ratio?
5. In the coordinate plane, right triangle $\triangle ABC$ is shown with $\angle C$ marked as a right angle. Point $A$ is to the left of $C$, and point $B$ is above $C$, so $\overline{AC}$ is horizontal and $\overline{BC}$ is vertical. The acute angle at $A$ is labeled $\theta$. (The diagram is not drawn to scale.) Which ratio represents $\sin(\theta)$?
6. A parabola is the set of points equidistant from the focus $F(-4,0)$ and the directrix $x=0$ (shown). The parabola opens to the left. Which equation represents the parabola?
7. Which statement correctly identifies the center and radius of the circle given by $x^2+y^2+2x-12y+29=0$?
8. A right circular cylinder is shown standing upright. A slicing plane cuts the cylinder perpendicular to the circular bases and passes through the cylinder's central axis (so the plane is vertical and contains the axis). Which shape results from the cross-section shown?
9. A student attempts to find $\tan(15°)$ by using $\tan(45° - 30°)$ and the formula $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$. Given that $\tan(45°) = 1$ and $\tan(30°) = \frac{\sqrt{3}}{3}$, which calculation should the student perform?
10. In the right triangle $\triangle MNO$ shown, $\angle N$ is a right angle and the hypotenuse is $\overline{MO}$. If $MN=7$ and $NO=24$, what is the length of $\overline{MO}$?
11. In the right triangle $\triangle ABC$ shown, $\angle C$ is a right angle (marked). The acute angles are labeled $\angle A = \theta$ and $\angle B = \varphi$, so $\theta$ and $\varphi$ are complementary. Which statement correctly relates $\sin(\theta)$ and $\cos(\varphi)$?
12. Using the identity $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$, if $x = 5$ and $y = 12$, which Pythagorean triple is generated?
13. Point $R$ is reflected across line $m$ to produce point $R'$. Line $m$ has equation $y = 2x + 1$. Which condition must the line $\overline{RR'}$ satisfy based on the definition of reflection?
14. An ice cream cone is a right circular cone with radius $4\text{ cm}$ and height $12\text{ cm}$. Which calculation correctly applies the volume formula for the cone?
15. In triangle $ABC$, angle $C$ is a right angle. If $\sin A = \frac{5}{13}$ and $AB = 26$, what is the length of side $BC$?
16. Triangle $RST$ has vertices $R(-3,1)$, $S(1,5)$, and $T(5,0)$ in the coordinate plane (units are uniform). What is the area of the triangle?
17. In right triangle $RST$ with right angle at $T$, $\sin R = \frac{12}{13}$ and $RS = 39$. Triangle $RST$ is similar to triangle $UVW$ where $UV = 26$. What is $\cos V$ in triangle $UVW$?
18. Triangle $ABC$ has vertices $A(0, 0)$, $B(8, 0)$, and $C(0, 6)$. A line parallel to $BC$ intersects $AB$ at point $P(5, 0)$ and $AC$ at point $Q$. What are the coordinates of point $Q$?
19. A right triangle has legs of length $a$ and $b$ and hypotenuse of length $c$. A second triangle is formed by connecting the midpoints of the three sides of the original triangle. If the area of the original triangle is 60 square units, what is the area of the triangle formed by connecting the midpoints?
20. In the coordinate plane, triangle $ABC$ is dilated about center $O$ with scale factor $k=\tfrac{3}{2}$ to form triangle $A'B'C'$. How does the length of segment $\overline{AB}$ compare to the length of its image $\overline{A'B'}$?

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