Practice Test 8 - Geometry

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

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1. A parabola is defined as the set of points equidistant from the focus $F(3,-2)$ and the directrix $y=2$. The parabola opens downward. Which equation represents the parabola?
2. In the right triangle shown, $\angle C$ is a right angle and $\angle A=\theta$. Which reasoning proves $\sin^2\theta+\cos^2\theta=1$?​
3. Two circles have equations $(x-2)^2 + (y+1)^2 = 9$ and $(x+3)^2 + (y-4)^2 = 25$. To prove these circles are similar, which transformation sequence correctly maps the first circle to the second?
4. In the right triangle shown, $\angle A$ is a right angle. The hypotenuse is $\overline{BC}$. If $AB=8$ and $BC=17$, what is the length of $AC$?
5. A cylindrical can has diameter $10\text{ cm}$ and height $12\text{ cm}$. What is the volume of the can in cubic centimeters?
6. On the coordinate plane, segment $\overline{AB}$ has endpoints $A(0,2)$ and $B(8,-6)$. Point $P$ divides the directed segment from $A$ to $B$ internally in the ratio $AP:PB=5:3$. Which coordinates represent the partition point?
7. Circles $\odot R$ and $\odot S$ are shown with centers marked. The radii are labeled $\overline{RA}=6$ and $\overline{SB}=9$. Which transformation maps one circle to the other to show they are similar?
8. A transformation $T$ maps point $(2, 3)$ to $(5, 7)$ and point $(4, 1)$ to $(7, 5)$. If $T$ is a translation, what is the image of point $(0, -2)$ under this transformation?
9. A right circular cylinder stands upright. A plane slices the cylinder perpendicular to the circular bases and passes through the cylinder's central axis (so the plane contains the axis). Which description correctly identifies the resulting shape?
10. Triangle $ABC$ has vertices $A(1,1)$, $B(3,1)$, and $C(2,4)$. It is transformed by the matrix $K=\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}.$ What are the coordinates of $C'$ (the image of $C$)?​
11. A triangle $\triangle PQR$ is shown, and a second triangle $\triangle STU$ appears to be a rotated and dilated image of it. Angle markings show $\angle P$ and $\angle S$ each have a single arc, and $\angle Q$ and $\angle T$ each have a double arc. No side lengths are labeled, and the diagram is not drawn to scale. Which reasoning uses similarity transformations correctly?
12. A right triangle has legs of length $a$ and $b$, and the angle opposite leg $a$ measures $\theta$. If the area of this triangle is 24 square units and $\tan \theta = \frac{3}{4}$, what is the length of the hypotenuse?
13. A hyperbola has foci at $F_1(0,-6)$ and $F_2(0,6)$. For any point $P(x,y)$ on the hyperbola, $|PF_2-PF_1|=8$. Which equation represents the hyperbola? (No asymptotes are given.)
14. A student attempts to construct a tangent line from external point $R$ to circle $C$ using compass and straightedge. The construction involves drawing a circle with diameter $RC$ where $C$ is the center of the original circle. Which statement best explains why this construction method works?
15. A teacher demonstrates Cavalieri's principle for sphere volume by slicing a hemisphere and a cylinder with a cone removed. Refer to the figure. If both solids are sliced by planes parallel to their bases, what must be true about the relationship between areas $A_1$ and $A_2$?
16. A lighthouse beam makes an angle of depression $\theta$ from the horizontal to a boat. The lighthouse light is 48 m above sea level, and the boat is 90 m horizontally from the point directly below the light. Which angle represents the correct solution within $0^\circ<\theta<90^\circ$? Solve: $\tan(\theta)=\dfrac{48}{90}$.
17. A cylindrical water tank has radius $3\text{ m}$ and height $10\text{ m}$. What is the volume of the tank in cubic meters?
18. In right triangle $\triangle QRS$ shown, $\angle R$ is a right angle. The legs are $QR=12$ and $RS=5$. What is the length of hypotenuse $QS$?
19. A transformation is represented by the matrix $A=\begin{pmatrix}2&0\\0&3\end{pmatrix}$. Which claim about area scaling is correct for a square region in the plane?
20. Two triangles $\triangle RST$ and $\triangle XYZ$ are shown in the plane. Angle markings show $\angle R$ and $\angle X$ each have a single arc, and $\angle S$ and $\angle Y$ each have a double arc. No side lengths are given, and the diagram is not drawn to scale. Which claim about side lengths must be true?

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