is a right triangle with right angle at . If and , what is the length of , in units?

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SAT Right Triangles & Trigonometry Question 38 | Free SAT Prep | Aniko

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Question 38·Medium·Right Triangles and Trigonometry

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$\triangle ABC$ is a right triangle with right angle at $C$ . If $\sin A = \dfrac{3}{5}$ and $AB = 20$ , what is the length of $AC$ , in units?

For right-triangle trigonometry questions, first locate the right angle to identify the hypotenuse. Then, for a given angle, clearly mark which side is opposite and which is adjacent. Use the appropriate trig ratio (here, sine = opposite/hypotenuse) to find one missing side. Once you know one leg and the hypotenuse, quickly apply the Pythagorean theorem to get the other leg. When the sine ratio uses a familiar fraction like 3/5, recognize the 3–4–5 right triangle pattern: the other leg must be in the 4-part of that ratio, scaled to match the given hypotenuse, which lets you move even faster.

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Identify the hypotenuse

The right angle is at C. In any right triangle, the hypotenuse is the side opposite the right angle. Which side is that here?

Use the definition of sine

For angle A, which side is opposite and which side is the hypotenuse? Write sin A as (opposite side) divided by (hypotenuse), then plug in sin A = 3/5 and AB = 20.

Apply the Pythagorean theorem

Once you find the length of the side opposite angle A, use the Pythagorean theorem, $\text{(leg)}^2 + \text{(leg)}^2 = \text{(hypotenuse)}^2$ , to solve for AC.

Desmos Guide

Compute the opposite leg using the sine ratio

In the first expression line, type 20*(3/5) and look at the output. This value is the length of the side opposite angle A (BC).

Use the Pythagorean theorem to find AC

In a new expression line, type sqrt(20^2 - (20*(3/5))^2). The numerical result Desmos shows for this expression is the length of AC.

Step-by-step Explanation

Identify the sides and use the sine ratio

The right angle is at C, so side AB is the hypotenuse.

For angle A:

The side opposite A is BC.
The hypotenuse is AB.

We are given sin A = 3/5 and AB = 20. By definition of sine:

sin A = (opposite) / (hypotenuse) = BC / AB.

\frac{BC}{20} = \frac{3}{5}

Multiply both sides by 20 to find BC:

BC = 20 \cdot \frac{3}{5} = 12.

Now you know one leg: BC = 12.

Set up the Pythagorean theorem to solve for AC

In right triangle ABC, the Pythagorean theorem says:

AC^2 + BC^2 = AB^2.

We know BC = 12 and AB = 20, so substitute these values:

AC^2 + 12^2 = 20^2.

Compute the squares:

AC^2 + 144 = 400.

Now isolate $AC^2$ .

Solve for AC

From

AC^2 + 144 = 400,

subtract 144 from both sides:

AC^2 = 400 - 144 = 256.

Since AC is a length, take the positive square root:

AC = 16.

So the length of $AC$ is 16 units.

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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