Practice Test 3 - Geometry

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

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1. The figure shows quadrilateral $ABCD$ and its image $A'B'C'D'$ under transformation $R$. What function rule best represents transformation $R$?
2. In the coordinate plane shown, triangle ABC has vertices at A(-2, 1), B(0, 4), and C(3, 1). Triangle DEF has vertices at D(1, -1), E(4, 2), and F(7, -1). A transformation T maps triangle ABC to triangle DEF. If $\angle BAC = \angle EDF = 63°$ and $\angle ABC = \angle DEF = 54°$, what can be determined about transformation T?
3. A parabola has the equation $(x - 4)^2 = -8(y + 2)$. What is the distance between the focus and the directrix of this parabola?
4. A right triangle has angle $\theta$ and side lengths: opposite $=a$, adjacent $=b$, hypotenuse $=c$. Which statement justifies the identity using the correct side ratios and the Pythagorean Theorem?
5. Triangle $PQR$ is inscribed in a circle with center $O$ and radius 10. If arc $PQ$ measures $120°$ and arc $QR$ measures $100°$, what is the measure of $\angle QPR$?
6. Two right triangles $\triangle GHI$ and $\triangle JKL$ are shown. Both have a right angle at $H$ and $K$ respectively, and both have an acute angle labeled $\theta$ at $G$ and $J$. Which statement correctly defines a trigonometric ratio using similarity?
7. Triangle $DEF$ is similar to triangle $GHI$ by the AA criterion, with $\angle D \cong \angle G$ and $\angle E \cong \angle H$. If triangle $DEF$ undergoes a sequence of transformations (reflection, then rotation, then dilation with scale factor $3$) to produce triangle $D''E''F''$, what can be concluded about the relationship between triangles $GHI$ and $D''E''F''$?
8. Triangle $ABC$ has vertices $A(-4, 1)$, $B(2, 5)$, and $C(0, -3)$. The perpendicular bisector of side $\overline{AB}$ intersects the perpendicular bisector of side $\overline{BC}$ at point $H$. What can be proven about point $H$?
9. Triangle $\triangle PQR$ is shown in the plane (not necessarily right). The side $\overline{PQ}$ is labeled $a$ and the side $\overline{PR}$ is labeled $b$. The included angle at $P$ is labeled $C$ (so $C=\angle QPR$). A dashed altitude from $R$ meets $\overline{PQ}$ at $S$, with $\overline{RS}\perp\overline{PQ}$. Which expression uses the included angle correctly to give the area of $\triangle PQR$?
10. A metal soda can is modeled as a cylinder to estimate the amount of aluminum needed for the can. The model ignores the small rim at the top, the slight indentation at the bottom, and the thickness of the metal. (The model is not exact.) Which claim about the model is NOT justified?
11. Which symmetries does the polygon have? Consider rotations about the polygon's center and reflections across lines in the plane.
12. On the coordinate plane, point $G(1,2)$ maps to $G'(2,1)$ and point $H(4,-1)$ maps to $H'(-1,4)$ under transformation $T$. Which description correctly represents this transformation as a function?
13. Given that quadrilateral $ABCD$ has vertices $A(-2, 3)$, $B(4, 1)$, $C(2, -3)$, and $D(-4, -1)$, which theorem can be used to prove that $ABCD$ is a parallelogram?
14. A right circular cone has radius $6\text{ ft}$ and height $8\text{ ft}$. What is the volume of the cone in cubic feet?
15. A right triangle $\triangle JKL$ is shown with $\angle K$ explicitly marked as $90^\circ$. The acute angles are labeled $\theta=\angle J$ and $\varphi=\angle L$, so $\theta+\varphi=90^\circ$. Which relationship must be true for complementary angles?
16. In right triangle $\triangle RST$, the right angle is at $S$. One leg is $RS=10$, and $\angle R = 28^\circ$. What is the length of hypotenuse $RT$?
17. A rectangle with side lengths $x+y$ and $x+z$ is partitioned by one vertical and one horizontal segment into four smaller rectangles with side lengths $x$, $y$, and $z$ as labeled. Which expression represents the total area (true for all $x$, $y$, and $z$)?
18. Using only a compass and straightedge, a student is constructing the angle bisector of $\angle ABC$. After drawing an arc centered at $B$ that intersects $\overrightarrow{BA}$ at $D$ and $\overrightarrow{BC}$ at $E$, the student draws arcs centered at $D$ and $E$ but chooses different compass widths for those two arcs. Which reasoning explains why the construction may fail?
19. In the diagram, $\triangle ABC$ is not a right triangle. The included angle at $A$ is labeled $\angle A=\theta$. The sides adjacent to $\angle A$ are labeled $AB=c$ and $AC=b$, and $BC=a$. Which setup correctly identifies the special case that connects the Law of Cosines to the Pythagorean Theorem?
20. An ellipse has center at the origin and passes through points $(5, 0)$ and $(0, 3)$. If the foci lie on the x-axis, what are the coordinates of the foci?

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