Practice Test 1 - Geometry

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

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1. A triangle has vertices $A(1,1)$, $B(3,1)$, and $C(2,3)$. The triangle is transformed by the matrix $S=\begin{pmatrix}2&0\\0&2\end{pmatrix}$. What are the coordinates of $C'$ (the image of $C$)?
2. A right circular cone is shown. A slicing plane passes through the cone's apex and also passes through the center of the circular base (so the plane contains the cone's axis). Which shape results from the cross-section shown?
3. A concrete pillar is a cylinder with diameter $12\text{ ft}$ and height $5\text{ ft}$. What is the volume of the solid?
4. Consider the informal argument for sphere volume using Cavalieri's principle with a hemisphere, cylinder, and cone. If the hemisphere has radius $R$, and we slice at distance $d$ from the center (where $0 \leq d \leq R$), which expression correctly represents the cross-sectional area of the hemisphere?
5. A dilation with center $C$ maps point $A$ to point $A'$ and point $B$ to point $B'$. Line $AB$ does not pass through $C$. If $\overrightarrow{CA} = (3, -2)$ and $\overrightarrow{CA'} = (9, -6)$, and line $AB$ has slope $\frac{3}{4}$, what is the slope of line $A'B'$?
6. Triangle $PQR$ with vertices $P(3, 4)$, $Q(7, 2)$, $R(5, 8)$ is congruent to triangle $STU$. If the congruence can be established through rigid motions, but triangle $STU$ has vertices $S(-4, 3)$, $T(-2, 7)$, $U(-8, 5)$, what must be verified to confirm the triangles are indeed congruent?
7. In the diagram, $ABC$ is not a right triangle. The included angle at $A$ is $\theta$, with adjacent sides $AB=c$ and $AC=b$, and opposite side $BC=a$. A student claims: "If $\theta=90^\circ$, then the Law of Cosines becomes the Pythagorean Theorem." Which expression correctly shows this special case?
8. In the diagram, $\triangle ABC$ is a right triangle with right angle at $C$. Segment $CD$ is drawn from $C$ to the hypotenuse $AB$, and $CD \perp AB$ at $D$. The legs are labeled $AC=b$ and $BC=a$, and the hypotenuse is labeled $AB=c$. Using triangle similarity created by the altitude (not memorization of a formula), which conclusion can be proven using the diagram?
9. In a certain city, the number of daylight hours varies sinusoidally over the year. The city has about 8 hours of daylight at its minimum (in December) and about 16 hours at its maximum (in June). The pattern repeats every 12 months. What are the amplitude and midline of a sinusoidal model for daylight hours?
10. A carpenter is sanding a wooden ball that is approximately the shape shown (not drawn to scale). The model ignores tiny scratches and treats the wood as perfectly round. The carpenter wants to estimate how much sandpaper is needed to sand the outside. Which property of the model is most relevant?
11. In the diagram, $\triangle ABC$ is right with $\angle A$ marked $90^\circ$. An altitude $AD$ is drawn to the hypotenuse $BC$ and is marked perpendicular at $D$. The hypotenuse is labeled $BC=c$ and is split into $BD=m$ and $DC=n$ (these are labels, not computed values). No other lengths are provided, and the diagram is not drawn to scale. Which relationship between side lengths must be true from triangle similarity?
12. In a right triangle, $\cos(\theta)=\frac{5}{13}$ and $0^\circ<\theta<90^\circ$. Which is $\theta$ (in degrees) to the nearest tenth?
13. A garden path is a rectangle $12\text{ ft}$ long and $8\text{ ft}$ wide. A rectangular flower bed must be placed inside it so that there is a uniform border of $1\text{ ft}$ of path on all sides. The flower bed's side lengths must be whole numbers of feet. Which design satisfies all constraints?
14. A line $p$ is tangent to circle $\odot Q$ at point J. Segment $QJ$ is drawn, and the right angle between $QJ$ and $p$ at $J$ is marked. Which reasoning correctly uses the radius–tangent relationship?
15. A student is constructing a line parallel to $\ell$ through point $P$ using only a compass and straightedge. The student correctly copies the angle at $X$ onto point $P$ using arcs and a copied chord, and then draws a line $m$ through $P$ forming the copied angle with the transversal. Which statement correctly describes the constructed line $m$?​
16. Two triangles, MNP and QRS, are being tested for similarity. It is known that $\angle M = \angle Q = 48°$ and that triangle QRS can be obtained from triangle MNP through a sequence of similarity transformations. A student concludes that this information alone is sufficient to prove similarity using the AA criterion. What is the error in this reasoning?
17. In the figure, triangles ABC and DEF are shown. It is given that $\angle A = 41°$, $\angle B = 67°$, $\angle D = 41°$, and $\angle F = 72°$. A student claims that since both triangles have an angle of 41°, and similarity transformations preserve angle measures, the triangles must be similar by the AA criterion. What is the primary flaw in this reasoning?
18. In the diagram, $\triangle ABC$ is shown in the plane. Segments $AB$ and $AC$ have matching single tick marks, indicating they are congruent. No angle arcs, parallel marks, right-angle boxes, midpoint markings, or lengths are given, and the diagram is not drawn to scale. Which statement must be true?
19. An ellipse has foci at $(-2, 1)$ and $(4, 1)$ and passes through the point $(1, 5)$. What is the value of $a^2$ in the standard form equation of this ellipse?
20. In the right triangle shown, angle $\theta$ is at vertex $A$. Which argument uses the Pythagorean Theorem correctly to prove that $\sin^2\theta+\cos^2\theta=1$ (for all $\theta$ where the triangle is defined)?

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