In right triangle , angle is a right angle. The length of is , and the measure of angle is . What is the length of ?

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SAT Right Triangles & Trigonometry Question 50 | Free SAT Prep | Aniko

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Question 50·Medium·Right Triangles and Trigonometry

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In right triangle $ABC$ , angle $C$ is a right angle. The length of $AC$ is $6\sqrt{3}$ , and the measure of angle $A$ is $30^\circ$ . What is the length of $BC$ ?

For right-triangle questions with angles of 30°, 60°, or 45°, first check whether the triangle is a special right triangle (30-60-90 or 45-45-90). Memorize the side ratios $1:\sqrt{3}:2$ for 30-60-90 and $1:1:\sqrt{2}$ for 45-45-90, and match the given side to the correct part of the ratio (short leg, long leg, or hypotenuse). This avoids using trigonometry and lets you solve quickly by simple proportional reasoning; if the angles are not special or you’re unsure, fall back on the appropriate trig function (sine, cosine, or tangent) based on which sides are given and which is unknown.

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Visualize and label the triangle

Sketch a right triangle and label the vertices A, B, and C so that angle C is the right angle. Then mark angle A as $30^\circ$ . Which angle must be $60^\circ$ ?

Identify which side is which

Remember that each side is opposite its angle. Which side is opposite $30^\circ$ ? Which side is opposite $60^\circ$ ? Which side is the hypotenuse?

Recall the special triangle ratio

A $30^\circ$ – $60^\circ$ – $90^\circ$ triangle has side lengths in the ratio $1 : \sqrt{3} : 2$ , corresponding to the sides opposite $30^\circ$ , $60^\circ$ , and $90^\circ$ . Match the given side $AC = 6\sqrt{3}$ to one of these parts and solve for the basic unit length.

Alternative approach with trigonometry

You can also use a trig function at angle A. Which side is opposite angle A and which side is adjacent to angle A? Which trig ratio compares opposite and adjacent sides?

Desmos Guide

Set up the trigonometric expression

In Desmos, enter the expression 6*sqrt(3)*tan(30 deg) (or switch Desmos to degree mode and enter 6*sqrt(3)*tan(30)). This comes from the equation $\tan 30^\circ = \frac{BC}{AC}$ , so $BC = AC \cdot \tan 30^\circ$ .

Interpret the result

Look at the numeric value that Desmos outputs for this expression. That value is the length of side $BC$ .

Step-by-step Explanation

Determine all the angles and identify the special triangle

We are told that angle C is a right angle, so $\angle C = 90^\circ$ . We are also told that $\angle A = 30^\circ$ .

In any triangle, the angles sum to $180^\circ$ , so

\angle A + \angle B + \angle C = 180^\circ

Substitute the known values:

30^\circ + \angle B + 90^\circ = 180^\circ

So $\angle B = 60^\circ$ . The triangle is therefore a $30^\circ$ – $60^\circ$ – $90^\circ$ triangle, a special right triangle.

Match each side to its angle

In any triangle, each side lies opposite its corresponding angle:

Side $BC$ is opposite angle $A$ .
Side $AC$ is opposite angle $B$ .
Side $AB$ is opposite angle $C$ (the right angle), so $AB$ is the hypotenuse.

We know $\angle A = 30^\circ$ and $\angle B = 60^\circ$ , so:

$BC$ is the side opposite $30^\circ$ (the shorter leg).
$AC$ is the side opposite $60^\circ$ (the longer leg). We are given $AC = 6\sqrt{3}$ .

Use the 30-60-90 triangle side ratios

In a $30^\circ$ – $60^\circ$ – $90^\circ$ triangle, the side lengths always follow this ratio:

Opposite $30^\circ$ : $x$ (shortest side)
Opposite $60^\circ$ : $x\sqrt{3}$ (longer leg)
Opposite $90^\circ$ : $2x$ (hypotenuse)

Here, $BC$ is opposite $30^\circ$ , so $BC$ corresponds to $x$ . The given side $AC$ is opposite $60^\circ$ , so $AC$ corresponds to $x\sqrt{3}$ .

We are told $AC = 6\sqrt{3}$ , so set up the equation:

x\sqrt{3} = 6\sqrt{3}

Solve for $x$ by dividing both sides by $\sqrt{3}$ :

x = 6

State the length of BC

From the 30-60-90 ratio, we defined $x$ as the side opposite $30^\circ$ , which is side $BC$ .

We found $x = 6$ , so the length of $BC$ is $6$ .

Therefore, the correct answer is 6.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation