Practice Test 11 - Geometry

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

Start practice test

1. A metal rod is modeled as a cylinder with radius $2\text{ cm}$ and height $15\text{ cm}$. The metal has density $7.8\ \text{g/cm}^3$. Which calculation gives the mass of the rod?
2. A class is discussing why the volume of a sphere of radius $r$ can be justified without calculus by comparing it to a cylinder of radius $r$ and height $2r$ with two congruent cones removed (each cone has base radius $r$ and height $r$). Which statement justifies the sphere's volume formula?
3. In the right triangle shown, the acute angle at $A$ is $\theta$. The hypotenuse is labeled $1$, the leg adjacent to $\theta$ is labeled $\cos\theta$, and the leg opposite $\theta$ is labeled $\sin\theta$. Which reasoning proves $\sin^2\theta+\cos^2\theta=1$?
4. A circle $\odot C$ is dilated about its center $C$ by a scale factor $k>0$ to produce a new circle $\odot C'$. The centers are marked. Which reasoning uses similarity correctly?
5. Points $E(-2,-1)$ and $F(4,11)$ are shown on a coordinate plane and connected by segment $\overline{EF}$. Point $R$ divides the directed segment from $E$ to $F$ internally in the ratio $ER:RF=3:1$. Which coordinates represent the partition point $R$?
6. Points $A(2, 5)$, $B(8, 1)$, $C(4, -5)$, and $D(-2, -1)$ form quadrilateral $ABCD$. Which statement can be proven using coordinate geometry?
7. A right triangle $\triangle RST$ is shown with the right angle marked at $S$. The hypotenuse is explicitly identified as $\overline{RT}$. The acute angle at $T$ is labeled $\theta$. No numeric lengths are given, and the diagram is not drawn to scale. Which ratio represents $\sin(\theta)$?
8. Triangle $DEF$ has vertices $D(-2,-1)$, $E(4,2)$, and $F(1,6)$ in the coordinate plane (units are uniform). Which calculation is required to find the area?
9. A party hat is modeled as a cone to estimate how much wrapping paper is needed to cover the outside. The model ignores the seam overlap, the thickness of the paper, and any brim. (The model is not exact.) Which property of the model is most relevant for answering the question about how much paper is needed to cover the outside?
10. A parabola opening rightward has vertex at $(-2, 4)$ and focus at $(1, 4)$. What is the equation of this parabola?
11. A circle has the equation $x^2 + y^2 - 6x + 8y - 11 = 0$. What is the equation of the circle that has the same radius but is centered at the origin?
12. A round fishbowl is a sphere with diameter $18\text{ cm}$. What is the volume of the solid?
13. On the coordinate plane, polygon $RSTU$ is mapped onto $R'S'T'U'$. Order matters. Which sequence correctly produces the image? $R(-1,1),\ S(2,1),\ T(2,2),\ U(-1,2)$ and $R'(5,4),\ S'(5,1),\ T'(4,1),\ U'(4,4)$.
14. Which matrix represents a reflection across the $y$-axis?​
15. Which transformation is represented by the matrix $T=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$?
16. The water level in a harbor is modeled by a sinusoidal function. The midline water level is 7 ft, and the amplitude is 3 ft. The time between consecutive high tides is 12 hours. Which choice gives the correct maximum and minimum water levels and the period?​
17. Triangle ABC has vertices at A(2, 3), B(8, 7), and C(5, 11). If the triangle is reflected across the y-axis to form triangle A'B'C', what is the difference between the perimeter of triangle ABC and the perimeter of triangle A'B'C'?
18. A linear transformation is represented by $G=\begin{pmatrix}1&1\\1&1\end{pmatrix}.$ The parallelogram formed by the vectors $\vec a=(1,0)$ and $\vec b=(0,1)$ (the unit square) is shown on the coordinate plane. Which conclusion follows from the determinant value?
19. A circle is cut into many equal sectors and then rearranged by alternating the sectors up and down to form a shape that looks like a parallelogram (not drawn to scale). The curved edges become the top and bottom boundaries, and the straight radii form the left and right sides. Which reasoning explains why the area formula for a circle is $A=\pi r^2$?
20. What is the domain of $\arcsin(x)$?

Practice tests