Practice Test 6 - Geometry

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

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1. In right triangle $\triangle DEF$ shown, $\angle E$ is a right angle. The hypotenuse is $DF=13$ and one leg is $DE=5$. What is the length of leg $EF$?
2. A linear transformation $T$ has matrix $M=\begin{pmatrix}1&0\\0&1\end{pmatrix}.$ Which reasoning correctly interprets the matrix?
3. A satellite dish is designed so that its cross-section forms a parabola. The receiver is placed at the focus, which is 6 inches from the vertex. If the vertex is at the origin and the dish opens upward, what is the equation of the parabolic cross-section?
4. The diagram shows a right triangle $\triangle JKL$ with $\angle K$ marked as a right angle. The acute angles are labeled $\angle J=\theta$ and $\angle L=\varphi$ (so they are complementary). Which identity follows from the diagram?
5. In triangle $ABC$, point $D$ is on side $AB$ and point $E$ is on side $AC$ such that $DE \parallel BC$. If $AD = 6$, $DB = 9$, and $AE = 8$, what is the perimeter of triangle $ADE$ given that the perimeter of triangle $ABC$ is 45?
6. In the coordinate plane, endpoints are $A(-3,6)$ and $B(9,0)$. Which coordinates represent the point $P$ that divides $\overline{AB}$ internally in the ratio $AP:PB=5:1$?
7. In right triangle $\triangle XYZ$, the right angle is at $Y$. The acute angles $\angle X$ and $\angle Z$ are labeled. The hypotenuse is the side opposite the right angle. Which ratio represents $\sin(\angle Z)$?
8. A matrix $A$ sends the basis vectors $\vec e_1=(1,0)$ and $\vec e_2=(0,1)$ to the points shown on the coordinate plane. Which statement describes the geometric effect of $A$ on area? Matrix: $A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$
9. Which statement correctly identifies the center and radius of the circle given by $x^2 + y^2 + 4x - 10y + 13 = 0$? Rewrite the equation in standard form to interpret it geometrically.
10. In a certain city, the number of daylight hours varies sinusoidally throughout the year. The city has about 8 hours of daylight at its minimum and about 16 hours at its maximum. The cycle repeats every 12 months. Which choice gives the correct amplitude, midline, and period for a sinusoidal daylight model?
11. Rectangle $LMNO$ is dilated about center $O$ with scale factor $k=\tfrac{4}{3}$ to form $L'M'N'O'$. Which description uses the scale factor correctly?
12. A right triangle has legs in the ratio $3:4$ and hypotenuse of length $h$. If the triangle is scaled by a factor of $k$ such that the area becomes $\frac{3h^2}{8}$, what is the value of $k$?
13. Two right triangles $\triangle DEF$ and $\triangle D'E'F'$ are shown. Each has a right angle at $E$ and $E'$ respectively, and each has an acute angle labeled $\theta$ at $D$ and $D'$. Which ratio is invariant under similarity (depends only on $\theta$) in these triangles?
14. A linear transformation $T$ in the plane is represented by the matrix $A=\begin{pmatrix}2&0\\0&\tfrac{1}{2}\end{pmatrix}.$ A rectangle in the coordinate plane has one corner at the origin and adjacent sides along the positive axes, with vertices $(0,0)$, $(4,0)$, $(4,2)$, and $(0,2)$. Which claim about area scaling is correct?
15. Which reasoning supports the angle addition formula? In the diagram, triangle $ABC$ has point $D$ on segment $AC$. At vertex $A$, ray $AD$ splits $\angle BAC$ into two angles labeled $\theta$ (between $AB$ and $AD$) and $\varphi$ (between $AD$ and $AC$), so $\angle BAC=\theta+\varphi$. Segment $BD$ is drawn. A right-angle marker indicates $BD\perp AC$ at $D$. Which statement proves an identity for $\sin(\theta+\varphi)$ using this construction?
16. Which expression represents all points $(x,y)$ that are 6 units from the center $(1,-3)$? (Use the distance formula connection.)
17. In the right triangle shown, $\angle B$ is a right angle. The hypotenuse is $\overline{AC}$. If $AB=9$ and $BC=12$, what is the length of $AC$? Give your answer in simplest form.
18. A shipping container is modeled as a rectangular prism with volume $12\ \text{m}^3$. A packing material fills it with energy density $250\ \text{J/m}^3$. Which claim is NOT justified by the density given?
19. A point $Q(2,5)$ is scaled by factor $3$ about the origin using the matrix $S=\begin{pmatrix}3&0\\0&3\end{pmatrix}.$ What is the image point $Q'$?
20. On the coordinate plane, segment $\overline{CD}$ has endpoints $C(1,5)$ and $D(9,1)$. Point $Q$ divides the directed segment from $C$ to $D$ internally in the ratio $CQ:QD=1:3$. Which point divides the segment in the given ratio?

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