Practice Test 10 - Geometry

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

Start practice test

1. What is the equation of the circle with center $(-2,-1)$ and radius $3$? (Justify using that all points on the circle are the same distance from the center.)
2. Which solid is formed by rotating a right triangle about one of its legs (that leg is the axis of rotation)?
3. In right triangle $\triangle GHI$ shown, $\angle H$ is a right angle. The leg $GH=4$ and hypotenuse $GI=9$. What is the measure of $\angle G$ (nearest degree)?
4. A point $P(5,2)$ is transformed by $T=\begin{pmatrix}-1&0\\ 0&1\end{pmatrix}.$ What is the image point $P'$?
5. A cube is shown. A slicing plane cuts the cube so that it is perpendicular to the top face and parallel to one pair of opposite side faces. Which shape results from the cross-section shown?
6. A company is cutting a circular logo (radius 5 cm) from a square sticker sheet. The logo must fit entirely inside the square, and the square's side length must be an integer number of centimeters to match a cutting template. Which square side length best meets the geometric requirements?
7. A pilot flies from airport A to airport B (150 miles away) and then to airport C. The angle at airport B is $125°$, and the distance from B to C is 200 miles. To find the direct distance from A to C using the Law of Cosines, what is the correct setup?
8. Circle $\odot M$ has center $M$ and Circle $\odot N$ has center $N$. The circles have different radii, and the centers are marked. Which statement explains why the circles are similar without using circumference or area formulas?
9. Triangle LMN has vertices $L(-2,0)$, $M(2,0)$, and $N(1,3)$ on the coordinate plane. Which value correctly represents the area?
10. In the diagram, $\triangle PQR$ is a right triangle with $\angle Q$ marked as $90^\circ$. The acute angles are labeled $\angle P=\theta$ and $\angle R=\varphi$ (so they are complementary). Which identity follows from the diagram (using right-triangle definitions and the fact that the two acute angles are complementary)?
11. Two right triangles $\triangle ABC$ and $\triangle A'B'C'$ are shown. Each has a right angle at $C$ and $C'$ respectively, and $\angle A \cong \angle A'$. Which relationship depends only on the angle and not on the size of the triangle?
12. A small museum is installing a rectangular display case inside a rectangular alcove that is $9\text{ ft}$ wide and $6\text{ ft}$ deep. Building code requires a clear walkway of at least $1\text{ ft}$ along each wall of the alcove (left, right, top, and bottom). Which design satisfies all constraints?
13. Rhombus $DEFG$ has vertices $D(1, 2)$, $E(4, 6)$, $F(8, 3)$, and $G(5, -1)$. To verify this quadrilateral is indeed a rhombus using coordinate geometry, which two properties must be proven?
14. Two solids are compared: (1) a sphere of radius $r$, and (2) a cylinder of radius $r$ and height $2r$ with two congruent cones removed (each cone has base radius $r$ and height $r$, with tips meeting at the center). Slices are taken by horizontal planes at height $h$ from the center. Which claim is NOT supported by the slicing argument used with Cavalieri's principle?
15. Triangle $PQR$ is shown in the plane. Point $M$ lies on segment $PQ$ and point $N$ lies on segment $PR$. The diagram marks $PM \cong MQ$ (matching tick marks on the two parts of $PQ$) and $PN \cong NR$ (matching tick marks on the two parts of $PR$). Segment $MN$ is drawn. No angle measures, no parallel markings, and no lengths are given, and the diagram is not drawn to scale. Which statement must be true?
16. A city park is modeled as a rectangle $1.5$ miles long and $0.8$ miles wide. The park has a population density of $2{,}400$ people per square mile during a festival. Which calculation gives the total number of people in the park?
17. Triangle ABC has angles measuring 45°, 60°, and 75°. Triangle DEF undergoes a sequence of similarity transformations (rotation, reflection, and dilation with scale factor 3) to produce triangle GHI. If triangle GHI has angles measuring 45°, 60°, and 75°, which statement about the relationship between triangles ABC and GHI is most accurate?
18. A company is cutting a circular logo (radius 5 cm) from a square sticker sheet. The logo must fit entirely inside the square, and the square's side length must be an integer number of centimeters to match a cutting template. Which square side length best meets the geometric requirements?​
19. Quadrilateral $RSTU$ has vertices $R(-3,-2)$, $S(1,0)$, $T(3,-4)$, and $U(-1,-6)$. A student claims $RSTU$ is a parallelogram. Which argument correctly uses coordinate geometry?
20. A party cone is filled with candy. The cone has radius $6\text{ cm}$ and height $10\text{ cm}$. Which calculation correctly applies the volume formula?

Practice tests