Question

determining the side length ratio for 30°-60°-90° triangles
which 30°-60°-90° triangle is labeled with the correct side length ratio?
(three triangles are shown, with side and angle labels: first triangle has right angle, 30°, 60°, sides 1, √3, 2; second triangle has right angle, 30°, 60°, sides 1, √3, 2; third triangle has right angle, 30°, 60°, sides 1, √3, 2 but with different orientation)

Explanation:

Step1: Recall 30-60-90 triangle rules

In a $30^\circ$-$60^\circ$-$90^\circ$ right triangle:

• Side opposite $30^\circ$: shortest side $= s$
• Side opposite $60^\circ$: $s\sqrt{3}$
• Hypotenuse (opposite $90^\circ$): $2s$

Step2: Verify first triangle

Opposite $30^\circ$: $\sqrt{3}$, opposite $60^\circ$: $1$, hypotenuse: $2$. This violates the rule (shortest side should be opposite $30^\circ$).

Step3: Verify second triangle

Opposite $30^\circ$: $1$, opposite $60^\circ$: $\sqrt{3}$, hypotenuse: $2$. This matches $s=1$, $s\sqrt{3}=\sqrt{3}$, $2s=2$.

Step4: Verify third triangle

Opposite $30^\circ$: $\sqrt{3}$, opposite $60^\circ$: $1$, hypotenuse: $2$. This violates the rule (shortest side not opposite $30^\circ$).

Answer:

The middle triangle (with $30^\circ$ opposite side length 1, $60^\circ$ opposite side length $\sqrt{3}$, hypotenuse length 2) is correctly labeled.