In right triangle , angle is the right angle. If and the length of the leg adjacent to angle is centimeters, what is the length of the leg opposite angle , in centimeters?

Solution available with step-by-step explanation

SAT Right Triangles & Trigonometry Question 37 | Free SAT Prep | Aniko

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Question 37·Easy·Right Triangles and Trigonometry

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In right triangle $XYZ$ , angle $Y$ is the right angle. If $\tan X = \tfrac{3}{4}$ and the length of the leg adjacent to angle $X$ is $8$ centimeters, what is the length of the leg opposite angle $X$ , in centimeters?

For right-triangle trigonometry questions, immediately recall the definitions: $\tan = \tfrac{\text{opposite}}{\text{adjacent}}$ , $\frac{\text{opp}}{\text{adj}}$ . Translate the trig statement into a simple fraction equation using a variable for the unknown side, then solve the resulting proportion with quick cross-multiplication rather than computing any trig functions on a calculator. Always double-check that you matched “opposite” and “adjacent” to the correct angle named in the problem.

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Recall what tangent represents

Think about the definition of $\tan X$ in a right triangle. Which sides of the triangle (relative to angle $X$ ) does it compare?

Set up a ratio with a variable

Let the opposite side to angle $X$ be $o$ . Write an equation using $\tan X = \tfrac{3}{4}$ and the fact that the adjacent side is $8$ .

Use a proportion to solve for the unknown

Your equation should look like a fraction with $o$ on top and $8$ on the bottom, equal to $\tfrac{3}{4}$ . How can you solve that proportion for $o$ ?

Desmos Guide

Compute the proportion directly

In Desmos, type 8 * (3/4) and look at the output. This value is the length of the leg opposite angle $X$ that corresponds to the tangent ratio $\tfrac{3}{4}$ with an adjacent side of 8.

Step-by-step Explanation

Use the definition of tangent

In a right triangle, for an acute angle $X$ ,

\tan X = \frac{\text{opposite leg}}{\text{adjacent leg}}.

You are told that $\tan X = \tfrac{3}{4}$ , so the ratio of the opposite leg to the adjacent leg at angle $X$ is $3:4$ .

Relate the given side to the 3:4 ratio

Let the leg opposite angle $X$ be $o$ , and the adjacent leg is given as $8$ .

Using the tangent ratio,

\frac{o}{8} = \frac{3}{4}.

This sets up a proportion between the actual triangle and the $3:4$ ratio triangle.

Solve the proportion for the opposite leg

Solve

\frac{o}{8} = \frac{3}{4}

by cross-multiplying:

4o = 3 \cdot 8 = 24.

Now divide both sides by $4$ :

o = \frac{24}{4} = 6.

So, the length of the leg opposite angle $X$ is $6$ centimeters.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation