Question

which $30^\circ$-$60^\circ$-$90^\circ$ triangle is labeled with the correct side length ratio?

Explanation:

Step1: Recall 30-60-90 triangle rules

In a 30°-60°-90° right triangle, the side lengths follow the ratio:

• Side opposite 30°: $s$
• Side opposite 60°: $s\sqrt{3}$
• Hypotenuse (opposite 90°): $2s$

where $s$ is the length of the shortest side.

Step2: Analyze first triangle

Right angle, 60° at top, 30° at bottom.

• Side opposite 30°: $\sqrt{3}$
• Side opposite 60°: $1$
• Hypotenuse: $2$

This violates the ratio, since the side opposite 30° should be the shortest side.

Step3: Analyze second triangle

Right angle at top, 30° at left, 60° at right.

• Side opposite 30°: $\sqrt{3}$
• Side opposite 60°: $1$
• Hypotenuse: $2$

This violates the ratio, as the side opposite 30° is longer than the side opposite 60°.

Step4: Analyze first triangle (corrected check)

Wait, recheck first triangle:

• Side opposite 30°: $\sqrt{3}$ (incorrect, should be shortest)
• Side opposite 60°: $1$ (shorter than 30° side, wrong)

Step5: Recheck second triangle

• Side opposite 30°: $\sqrt{3}$ (wrong, should be shortest)
• Side opposite 60°: $1$ (shorter, wrong)

Step6: Recheck first triangle correctly

First triangle:

• Right angle at bottom left, 60° at top left, 30° at bottom right.
• Side opposite 30°: $1$ (correct, shortest side)
• Side opposite 60°: $\sqrt{3}$ (correct, $1 \times \sqrt{3}$)
• Hypotenuse: $2$ (correct, $1 \times 2$)

This matches the 30-60-90 ratio.

Answer:

The first triangle (leftmost triangle: right angle at bottom left, 60° at top left, 30° at bottom right, with sides 1, $\sqrt{3}$, 2) is labeled correctly.