Practice Test 5 - Geometry

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

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1. Two students try to justify the sphere volume formula by comparing a sphere of radius $r$ to a cylinder of radius $r$ and height $2r$ minus two cones. Student 1 takes slices at height $h$ measured from the cylinder's bottom. Student 2 takes slices at height $h$ measured from the cylinder's center. Which comparison uses Cavalieri's principle correctly?
2. Rhombus $DEFG$ has vertices at $D(0,0)$, $E(3,4)$, $F(8,4)$, and $G(5,0)$. The diagonals of the rhombus intersect at point $H$. Which transformation carries the rhombus onto itself?
3. An ellipse has foci at $F_1(-3, 0)$ and $F_2(3, 0)$. If the point $(0, 4)$ lies on the ellipse, what is the equation of the ellipse in standard form?
4. A point on a rotating wheel has its height above the ground modeled by a sinusoidal function. The height ranges from 5 ft (minimum) to 75 ft (maximum), and one full rotation takes 10 minutes. What are the amplitude, midline, and period?
5. A solid metal sphere has radius $6\text{ cm}$. Which calculation correctly applies the volume formula?
6. A right circular cylinder is shown standing upright. A slicing plane cuts the cylinder parallel to the circular bases (so the plane is horizontal). Which shape results from the cross-section shown?
7. A circular fountain of radius 6 m is centered at the origin on a coordinate grid. A straight footbridge is to be built as a chord of the circle. City rules require the bridge to be at least 10 m long but not longer than 11 m. Which proposed bridge endpoints meet all constraints?
8. Triangle JKL undergoes a similarity transformation consisting of a reflection across line m followed by a dilation with scale factor k > 0. The resulting triangle J'K'L' has $\angle J' = 42°$ and $\angle K' = 85°$. If a student wants to prove that triangles JKL and J'K'L' are similar using the AA criterion, what information about triangle JKL is sufficient?
9. Consider a sphere of radius $r$ and a cylinder of radius $r$ and height $2r$ with two cones removed (each cone has height $r$ and base radius $r$). The solids are sliced by horizontal planes at height $h$ from the center. Which reasoning matches the cross-sections shown and correctly applies Cavalieri's principle?
10. A student claims that $\sin(x + y) + \sin(x - y) = 2\sin x \cos y$. To verify this identity, which approach would be most direct?
11. An ellipse has foci at $F_1(-1,4)$ and $F_2(7,4)$. For any point $P(x,y)$ on the ellipse, $PF_1+PF_2=18$. Which equation follows from the focus definition? (No directrix is given.)
12. A hyperbola has foci at $F_1(-6,-1)$ and $F_2(2,-1)$. For any point $P(x,y)$ on the hyperbola, $|PF_2-PF_1|=6$. Which equation represents the hyperbola? (No asymptotes are given.)
13. Two concentric circles are shown with the same center $O$. The smaller circle has radius $5$ and the larger circle has radius $20$. Which statement explains why the circles are similar?
14. A right triangle $\triangle ABC$ has right angle at $C$ and angle $\theta$ at $A$. The diagram labels $\sin\theta=\dfrac{BC}{AB}$ and $\cos\theta=\dfrac{AC}{AB}$. Which argument uses the Pythagorean Theorem correctly? Image description (not drawn to scale): A right triangle with vertices $A$ (left), $B$ (upper-right), $C$ (lower-right). A right-angle box marks $\angle C=90^\circ$. The hypotenuse is $AB$. The legs are $AC$ (adjacent to $\theta$ at $A$) and $BC$ (opposite to $\theta$). An angle arc at $A$ labels $\theta$. No numerical lengths are given; only segment names are shown.
15. 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?​
16. A Ferris wheel's passenger height above the ground is modeled by a sinusoidal function. The minimum height is $6$ ft and the maximum height is $74$ ft. The wheel completes one rotation every $14$ minutes. What are the amplitude and period of the height function?
17. Two chords $AB$ and $CD$ intersect inside circle $O$ at point $E$. If $AE = 6$, $EB = 4$, and $CE = 8$, what is the length of $ED$?
18. 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 $YZ$?​
19. In the diagram, $\triangle PQR$ is shown. Point $M$ lies on segment $PQ$ and point $N$ lies on segment $PR$. The markings show $PM \cong MQ$ (matching tick marks on $PM$ and $MQ$) and $PN \cong NR$ (matching tick marks on $PN$ and $NR$). Segment $MN$ is drawn. No parallel arrows, angle markings, or lengths are given, and the diagram is not drawn to scale. Which conclusion follows from the diagram?
20. An ellipse has foci at $F_1(-1, 2)$ and $F_2(5, 2)$. If point $Q(2, 6)$ lies on the ellipse, what is the equation of the ellipse in standard form centered at its center?

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