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Geometry

Geometry Help: Trigonometric Ratios From Right Triangle Similarity

Review real example questions for Trigonometric Ratios From Right Triangle Similarity in Geometry.

Question 1

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?

$\frac{4}{5}$
$\frac{6}{5}$
$\frac{12}{15}$
$\frac{6}{10}$

Explanation: Trigonometric ratios are preserved in similar triangles because they depend only on the angles, not the side lengths. The scale factor affects the actual side lengths but not their ratios. Choice B incorrectly multiplies by the scale factor. Choice C shows the same ratio in different form but suggests scaling. Choice D incorrectly applies the inverse scale factor.

Question 2

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?

Explanation: Since the triangles are similar, the sine of the corresponding angle in the larger triangle is also 3/5. In the larger triangle, sin(angle) = opposite/hypotenuse = opposite/40 = 3/5. Therefore, opposite = 40 × (3/5) = 24. Choice A would be correct for the smaller triangle. Choice C incorrectly applies the 5:8 ratio to the sine value. Choice D results from using cosine instead of sine.

Question 3

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?

$\sin(\theta)=\dfrac{HI}{GI}$ in both triangles
$\sin(\theta)=\dfrac{GI}{HI}$ in both triangles
$\sin(\theta)=\dfrac{GH}{GI}$ in both triangles
$\sin(\theta)=\dfrac{HI}{GH}$ in both triangles

Explanation: This problem tests understanding of how similarity defines trigonometric ratios across different triangles. Right triangles with the same acute angle θ are similar because they share two angles (θ and 90°), making the third angle equal. For both triangles GHI and JKL with right angles at H and K respectively and angle θ at G and J, we identify sides relative to θ: in triangle GHI, opposite is HI, adjacent is GH, and hypotenuse is GI. The sine of θ equals opposite/hypotenuse = HI/GI in the first triangle, and by similarity, this same ratio holds in any right triangle with angle θ. This angle-dependence, not triangle-dependence, is what makes trigonometric functions well-defined. Option D incorrectly uses HI/GH, which would be tan(θ), not sin(θ). Always verify your ratio matches the correct trigonometric function definition.

Question 4

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 $\sin(\theta)$ ?

$\dfrac{AC}{AB}$
$\dfrac{AB}{BC}$
$\dfrac{BC}{AB}$
$\dfrac{BC}{AC}$

Explanation: The skill involves defining trigonometric ratios using the similarity of right triangles. Right triangles with the same acute angle are similar because they each have a 90-degree angle and share one acute angle, making the third angles equal by the angle sum in a triangle. In triangle ABC with right angle at C and angle θ at A, the side opposite θ is BC, the adjacent side is AC, and the hypotenuse is AB. The sine of θ is defined as the ratio of the opposite side to the hypotenuse, which is BC/AB. This ratio depends only on the measure of θ and is constant across similar triangles. A common misconception is to select BC/AC, which represents the tangent of θ instead of sine. To apply this to other problems, always start by labeling the sides as opposite, adjacent, and hypotenuse relative to the given angle.

Question 5

In triangle $ABC$ , angle $C$ is a right angle. If $\sin A = \frac{5}{13}$ and $AB = 26$ , what is the length of side $BC$ ?

Explanation: Since sin A = opposite/hypotenuse = BC/AB, we have BC/26 = 5/13. Therefore BC = 26 × (5/13) = 10. Choice B results from confusing sine with cosine. Choice C comes from using the Pythagorean theorem incorrectly. Choice D incorrectly assumes BC equals the hypotenuse.

Question 6

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 $UVW$ ?

$\frac{24}{26}$
$\frac{12}{13}$
$\frac{10}{26}$
$\frac{5}{13}$

Explanation: This problem tests your understanding of similar triangles and trigonometric ratios. When triangles are similar, corresponding angles are equal, which means their trigonometric ratios are identical. First, let's work with triangle

RST

. Since

\sin R = \frac{12}{13}

and

RS = 39

, we can find the side lengths. The sine ratio tells us that

\frac{ST}{RS} = \frac{12}{13}

, so

ST = 39 \times \frac{12}{13} = 36

. Using the Pythagorean theorem,

RT = \sqrt{39^2 - 36^2} = \sqrt{1521 - 1296} = 15

. Therefore,

\cos R = \frac{RT}{RS} = \frac{15}{39} = \frac{5}{13}

. Since the triangles are similar, corresponding angles have equal trigonometric ratios. If

UV = 26

corresponds to the hypotenuse

RS = 39

, then angle

V

corresponds to angle

R

. Therefore,

\cos V = \cos R = \frac{5}{13}

. Looking at the wrong answers: Choice A (

\frac{24}{26}

) incorrectly assumes you can scale the adjacent side directly by the ratio

\frac{26}{39}

. Choice B (

\frac{12}{13}

) confuses cosine with sine from the original triangle. Choice C (

\frac{10}{26}

) attempts to scale the adjacent side

15

\frac{26}{39}

but makes calculation errors. Strategy tip: In similar triangle problems, remember that corresponding angles have identical trigonometric ratios regardless of the triangles' sizes. Find the ratio in one triangle, then identify which angles correspond between the triangles.

Question 7

A right triangle $\triangle RST$ is drawn with a right angle at $S$ . Angle $T$ is acute and labeled. The hypotenuse is the side opposite the right angle.

Which ratio represents $\cos(\angle T)$ ?

$\dfrac{ST}{RT}$
$\dfrac{RT}{ST}$
$\dfrac{RS}{RT}$
$\dfrac{RS}{ST}$

Explanation: The skill involves defining trigonometric ratios using the similarity of right triangles that share the same acute angle. Right triangles with the same acute angle are similar because they have two angles in common—the right angle and the matching acute angle—making the third angles equal as well. For angle T in triangle RST with right angle at S, the opposite side is RS, the adjacent side is ST, and the hypotenuse is RT. The cosine of angle T is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. This ratio is constant for any right triangle with the same acute angle T because similarity ensures proportional corresponding sides, depending solely on the angle's measure. A common misconception is to use opposite over hypotenuse for cosine, which actually defines sine. To apply this correctly, always label the sides as opposite, adjacent, and hypotenuse relative to the given angle before selecting the appropriate ratio.

Question 8

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?

$\dfrac{BC}{AB}=\sin(\angle A)$
$AB-BC$ is the same in both triangles
$AB$ is the same in both triangles
$BC+AC$ is the same in both triangles

Explanation: This problem explores which relationships remain constant in similar right triangles. When triangles ABC and A'B'C' both have right angles at C and C' respectively, and angle A is congruent to angle A', the triangles are similar by AA similarity. In any right triangle with angle A, the sine of A equals the ratio of the opposite side to the hypotenuse, which is BC/AB. This ratio depends only on angle A, not on the triangle's size, because similar triangles have proportional sides. Options B, C, and D involve specific lengths or sums that change with triangle size and are not ratios. The key insight is that trigonometric ratios are defined through similarity to be angle-dependent but size-independent. Students often mistakenly think that individual side lengths or their differences remain constant, but only ratios of sides are preserved under similarity.

Question 9

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 $\sin(\theta)$ ?

$\dfrac{RS}{RT}$
$\dfrac{ST}{RT}$
$\dfrac{RT}{ST}$
$\dfrac{RS}{ST}$

Explanation: The skill involves defining trigonometric ratios using the similarity of right triangles. Right triangles that share an acute angle

\theta

are similar because they both have angles

\theta

90^\circ

, and

90^\circ - \theta

, satisfying the AA similarity criterion. In triangle RST with right angle at S and

\theta

at T, the side opposite

\theta

is RS, the adjacent side is ST, and the hypotenuse is RT. The sine of

\theta

is defined as the ratio of the opposite side to the hypotenuse, which is

\frac{RS}{RT}

. This ratio depends only on the measure of

\theta

, as similar right triangles have corresponding sides in proportion, making the ratio constant for a given

\theta

. A common misconception is to choose

\frac{ST}{RT}

for sine, which uses adjacent instead of opposite and actually defines cosine. To apply this in any right triangle, first label the sides as opposite, adjacent, and hypotenuse relative to the given angle.

Question 10

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 $\tan(\alpha)$ ?

$\dfrac{QR}{PQ}$
$\dfrac{PR}{PQ}$
$\dfrac{PQ}{PR}$
$\dfrac{PQ}{QR}$

Explanation: The skill involves defining trigonometric ratios using the similarity of right triangles. Right triangles with the same acute angle are similar because they each have a 90-degree angle and share one acute angle, making the third angles equal by the angle sum in a triangle. In triangle PQR with right angle at Q and angle α at P, the side opposite α is QR, the adjacent side is PQ, and the hypotenuse is PR. The tangent of α is defined as the ratio of the opposite side to the adjacent side, which is QR/PQ. This ratio depends only on the measure of α and is constant across similar triangles. A common misconception is to select PQ/QR, which is the cotangent of α instead of tangent. To apply this to other problems, always start by labeling the sides as opposite, adjacent, and hypotenuse relative to the given angle.

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Geometry

Geometry Help: Trigonometric Ratios From Right Triangle Similarity

Review real example questions for Trigonometric Ratios From Right Triangle Similarity in Geometry.

Question 1

$\frac{4}{5}$
$\frac{6}{5}$
$\frac{12}{15}$
$\frac{6}{10}$

Question 2

Question 3

$\sin(\theta)=\dfrac{HI}{GI}$ in both triangles
$\sin(\theta)=\dfrac{GI}{HI}$ in both triangles
$\sin(\theta)=\dfrac{GH}{GI}$ in both triangles
$\sin(\theta)=\dfrac{HI}{GH}$ in both triangles

Question 4

$\dfrac{AC}{AB}$
$\dfrac{AB}{BC}$
$\dfrac{BC}{AB}$
$\dfrac{BC}{AC}$

Question 5

In triangle $ABC$ , angle $C$ is a right angle. If $\sin A = \frac{5}{13}$ and $AB = 26$ , what is the length of side $BC$ ?

Question 6

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 $UVW$ ?

$\frac{24}{26}$
$\frac{12}{13}$
$\frac{10}{26}$
$\frac{5}{13}$

RST

. Since

\sin R = \frac{12}{13}

and

RS = 39

, we can find the side lengths. The sine ratio tells us that

\frac{ST}{RS} = \frac{12}{13}

, so

ST = 39 \times \frac{12}{13} = 36

. Using the Pythagorean theorem,

RT = \sqrt{39^2 - 36^2} = \sqrt{1521 - 1296} = 15

. Therefore,

\cos R = \frac{RT}{RS} = \frac{15}{39} = \frac{5}{13}

. Since the triangles are similar, corresponding angles have equal trigonometric ratios. If

UV = 26

corresponds to the hypotenuse

RS = 39

, then angle

V

corresponds to angle

R

. Therefore,

\cos V = \cos R = \frac{5}{13}

. Looking at the wrong answers: Choice A (

\frac{24}{26}

) incorrectly assumes you can scale the adjacent side directly by the ratio

\frac{26}{39}

. Choice B (

\frac{12}{13}

) confuses cosine with sine from the original triangle. Choice C (

\frac{10}{26}

) attempts to scale the adjacent side

15

\frac{26}{39}

Question 7

A right triangle $\triangle RST$ is drawn with a right angle at $S$ . Angle $T$ is acute and labeled. The hypotenuse is the side opposite the right angle.

Which ratio represents $\cos(\angle T)$ ?

$\dfrac{ST}{RT}$
$\dfrac{RT}{ST}$
$\dfrac{RS}{RT}$
$\dfrac{RS}{ST}$

Question 8

$\dfrac{BC}{AB}=\sin(\angle A)$
$AB-BC$ is the same in both triangles
$AB$ is the same in both triangles
$BC+AC$ is the same in both triangles

Question 9

$\dfrac{RS}{RT}$
$\dfrac{ST}{RT}$
$\dfrac{RT}{ST}$
$\dfrac{RS}{ST}$

Explanation: The skill involves defining trigonometric ratios using the similarity of right triangles. Right triangles that share an acute angle

\theta

are similar because they both have angles

\theta

90^\circ

, and

90^\circ - \theta

, satisfying the AA similarity criterion. In triangle RST with right angle at S and

\theta

at T, the side opposite

\theta

is RS, the adjacent side is ST, and the hypotenuse is RT. The sine of

\theta

is defined as the ratio of the opposite side to the hypotenuse, which is

\frac{RS}{RT}

. This ratio depends only on the measure of

\theta

, as similar right triangles have corresponding sides in proportion, making the ratio constant for a given

\theta

. A common misconception is to choose

\frac{ST}{RT}

Question 10

$\dfrac{QR}{PQ}$
$\dfrac{PR}{PQ}$
$\dfrac{PQ}{PR}$
$\dfrac{PQ}{QR}$

Explanation: The skill involves defining trigonometric ratios using the similarity of right triangles. Right triangles with the same acute angle are similar because they each have a 90-degree angle and share one acute angle, making the third angles equal by the angle sum in a triangle. In triangle PQR with right angle at Q and angle α at P, the side opposite α is QR, the adjacent side is PQ, and the hypotenuse is PR. The tangent of α is defined as the ratio of the opposite side to the adjacent side, which is QR/PQ. This ratio depends only on the measure of α and is constant across similar triangles. A common misconception is to select PQ/QR, which is the cotangent of α instead of tangent. To apply this to other problems, always start by labeling the sides as opposite, adjacent, and hypotenuse relative to the given angle.