Geometry Quiz: Trigonometric Ratios From Right Triangle Similarity

Two similar right triangles have a scale factor of 3:2. If the cosine of the acute angle in the smaller triangle is $\frac{4}{5}$, what is the cosine of the corresponding acute angle in the larger triangle?

Two similar right triangles have corresponding sides in the ratio 5:8. In the smaller triangle, one acute angle has a sine of $\frac{3}{5}$. If the hypotenuse of the larger triangle is 40, what is the length of the side opposite to the corresponding acute angle in the larger triangle?

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?

In the coordinate plane, right triangle $\triangle ABC$ is shown with $\angle C$ marked as a right angle. Point $A$ is to the left of $C$, and point $B$ is above $C$, so $\overline{AC}$ is horizontal and $\overline{BC}$ is vertical. The acute angle at $A$ is labeled $\theta$. (The diagram is not drawn to scale.) Which ratio represents $\sin(\theta)$?

In right triangle $RST$ with right angle at $T$, $\sin R = \frac{12}{13}$ and $RS = 39$. Triangle $RST$ is similar to triangle $UVW$ where $UV = 26$. What is $\cos V$ in triangle $UVW$?

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?

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 $\sin(\theta)$?

A right triangle $\triangle PQR$ is shown in the plane with $\angle Q$ marked as a right angle. The acute angles at $P$ and $R$ are labeled $\alpha$ and $\beta$, respectively. (The diagram is not drawn to scale.) Which ratio represents $\tan(\alpha)$?

In right triangle $XYZ$ with right angle at $Z$, $\cos X = \frac{5}{13}$. If triangle $XYZ$ is similar to triangle $MNP$ with a scale factor of 4:1, and $MN$ (corresponding to $XY$) has length 52, what is the length of $NP$ (corresponding to $YZ$)?

Two right triangles $\triangle DEF$ and $\triangle D'E'F'$ are drawn. Each has a marked right angle at $E$ and $E'$, and the acute angle at $D$ and $D'$ is labeled $\theta$. A student claims that because the triangles are different sizes, the value of $\sin(\theta)$ must be different in each triangle. Which relationship depends only on the angle $\theta$ (and not on the triangle size)?

Two right triangles $\triangle UVW$ and $\triangle U'V'W'$ are shown. Each has a right angle at $V$ and $V'$ respectively, and $\angle W \cong \angle W'$. Which ratio represents $\sin(\angle W)$ in both triangles?

A right triangle $\triangle KLM$ is drawn with a right angle at $L$. Angle $K$ is acute and labeled. The hypotenuse is the side opposite the right angle.

Which ratio is invariant under similarity and equals $\tan(\angle K)$?

Triangle $PQR$ is similar to triangle $STU$ with a ratio of similarity of 2:3. If $\cos P = 0.6$ in triangle $PQR$, and the hypotenuse of triangle $STU$ is 30, what is the length of the side adjacent to angle $S$ (the angle corresponding to angle $P$)?

A right triangle $\triangle DEF$ is drawn with a right angle at $E$. Angle $D$ is acute and labeled. The hypotenuse is the side opposite the right angle.

Which statement correctly defines a trigonometric ratio for $\angle D$?

Two right triangles $\triangle ABC$ and $\triangle A'B'C'$ are shown with right angles at $C$ and $C'$ and the same acute angle $\theta$ at $A$ and $A'$. Which relationship depends only on the angle $\theta$ (and not on the triangle size)?

A right triangle $\triangle LMN$ is drawn with the right angle marked at $M$. The hypotenuse is explicitly identified as $\overline{LN}$. The acute angles at $L$ and $N$ are labeled $\alpha$ and $\beta$, respectively. No side lengths are given, and the diagram is not drawn to scale. Which ratio represents $\sin(\beta)$?

A right triangle $\triangle DEF$ is drawn in the plane with the right angle marked at $E$. The hypotenuse is explicitly identified as $\overline{DF}$. The acute angle at $D$ is labeled $\theta$. No side lengths are shown, and the diagram is not drawn to scale. Which expression defines $\cos(\theta)$?

In right triangle $\triangle ABC$ (right angle marked at $C$), the acute angles at $A$ and $B$ are labeled $\alpha$ and $\beta$. The hypotenuse is explicitly identified as $\overline{AB}$. No side lengths are given, and the diagram is not drawn to scale. Which statement correctly defines a trigonometric ratio?

A right triangle $\triangle STU$ is drawn with $\angle T$ marked as a right angle. The acute angle at $S$ is labeled $\theta$. A student says, "$\sin(\theta)$ is the ratio of the longer leg to the shorter leg." Which claim about the ratio is NOT justified by right-triangle similarity and the definition of sine?

In right triangle $\triangle VWX$, $\angle W$ is marked as a right angle. The acute angles at $V$ and $X$ are labeled $\theta$ and $\phi$, respectively. (The diagram is not drawn to scale.) Which ratio represents $\cos(\theta)$?