Practice Test 9 - Geometry

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

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1. A right triangle has hypotenuse length $1$, and angle $\theta$ is one acute angle. Which reasoning proves $\sin^2\theta+\cos^2\theta=1$ using the Pythagorean Theorem?
2. A square with side length s is dilated by scale factor $\frac{2}{3}$. The resulting square is then dilated by scale factor m. If the final square has side length equal to $\frac{4}{3}$ times the original side length s, what is the value of m?
3. A rectangular poster is to be centered on a wall section that is $14\text{ ft}$ wide and $10\text{ ft}$ tall. A border of at least $2\text{ ft}$ must remain between the poster and each edge of the wall section. The poster must have perimeter at least $28\text{ ft}$. Which design satisfies all constraints?
4. An ellipse has foci at $F_1(0,-2)$ and $F_2(0,2)$. The sum of distances from any point $P(x,y)$ on the ellipse to the foci is $10$. Which equation represents the ellipse? (No vertices are given.)
5. Two circles $\odot G$ and $\odot H$ are drawn with different radii. Their centers are marked at $G$ and $H$. Which property guarantees circle similarity?
6. A cone and a cylinder have the same base radius and height. If the volume of the cylinder is $432\pi$ cubic centimeters, what is the volume of the cone?
7. A surveyor measures an angle of elevation of $\theta$ to the top of a building. If $\cos(\theta) = 0.8$, what is $\sin(90° - \theta)$?
8. A rotating beacon points from north toward east. The angle $\theta$ is measured from the north direction, and the eastward component of the light's direction is modeled by $\sin(\theta)$. If the beacon's eastward component must be $0.6$, the model is $\sin(\theta)=0.6.$ Which angle represents the correct solution within $0^\circ\le \theta\le 90^\circ$?
9. A transformation is given by $A=\begin{pmatrix}4&0\\0&0\end{pmatrix}.$ Which statement describes the geometric effect on the unit square (area 1)?
10. In the diagram of triangle $\triangle ABC$, sides $AB$ and $AC$ are labeled $b$ and $a$, and the included angle at $A$ is labeled $C$. A dashed altitude from $C$ drops perpendicularly to side $AB$ at point $D$. Which expression represents the area of the triangle?
11. A pendulum's horizontal displacement from center is modeled by $x(\theta)=0.8\sin(\theta)$ (meters), where $\theta$ is the phase angle. During one cycle, $0\le \theta\le 2\pi$. At what angle(s) is the displacement $x=0.4$ m? Solve: $0.8\sin(\theta)=0.4$.
12. A sphere of radius $r$ is compared with a "difference solid" formed by taking a cylinder of radius $r$ and height $2r$ and removing two congruent cones (each cone has radius $r$ and height $r$, tips meeting at the cylinder's midpoint). Slices are taken perpendicular to the axis at height $h$ from the midpoint. Which explanation correctly subtracts volumes using Cavalieri's principle?
13. A swimmer's distance from the left end of a pool changes periodically as they swim back and forth at a steady pace. Their distance ranges from $0$ m (left wall) to $25$ m (right wall) and back, repeating every $40$ seconds. What are the amplitude and midline of a sinusoidal model for the swimmer's distance from the left wall?
14. A solar panel installer needs to determine the optimal tilt angle for a rectangular panel on a flat roof. The panel is 2 meters wide and 1.5 meters long. When tilted at angle θ from horizontal, the panel's effective area for sun exposure is the projection of its surface perpendicular to the sun's rays. If the sun's rays make a 60° angle with the horizontal, what tilt angle θ maximizes the effective area?
15. A dilation followed by a rotation is used to map triangle $\triangle UVW$ to triangle $\triangle U'V'W'$. In the diagram, $\angle U \cong \angle U'$ is marked with one arc and $\angle V \cong \angle V'$ is marked with two arcs. No side lengths are labeled, and the diagram is not drawn to scale. Which conclusion about the triangles is valid?
16. Refer to the figure. Triangle DEF is the image of triangle ABC under a dilation. If the scale factor of the dilation is $\frac{5}{3}$, and DE = 20 units, what is the length of AB?
17. What is the range of $\arccos(x)$ (in degrees)?
18. Quadrilateral $ABCD$ has coordinates $A(-1,-1)$, $B(3,1)$, $C(5,5)$, and $D(1,3)$. A student claims $ABCD$ is a kite. Which reasoning proves the figure is a kite?
19. A transformation is represented by the matrix $Z=\begin{pmatrix}0&0\\0&0\end{pmatrix}.$ Which transformation does the matrix represent when applied to any vector, such as $\vec v=(3,-2)$?
20. Use the figure. A plane cuts through the triangular prism parallel to the triangular base but not passing through any vertices. If the cut is made at $\frac{2}{5}$ of the height from the base, how does the resulting cross-section compare to the original triangular base?

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