Question

If the hypotenuse of a 30-60-90 triangle is 8 cm, how long is the side opposite the 60-degree angle?

the side opposite the 60-degree angle is ____ cm

Answer 1

In a 30-60-90 triangle, the ratio of the lengths of the sides opposite the 30-degree angle, 60-degree angle, and the hypotenuse is 1:√3:2.

Given that the hypotenuse is 8 cm, we can find the length of the side opposite the 60-degree angle using the ratio:

1. The hypotenuse (2x) = 8 cm
\[
x = \frac{8}{2} = 4 \text{ cm}
\]

2. The side opposite the 60-degree angle (x√3) is calculated as follows:
\[
\text{Length} = 4\sqrt{3} \text{ cm}
\]

To get a numerical approximation for \( 4\sqrt{3} \):
\[
\sqrt{3} \approx 1.732
\]
Thus,
\[
4\sqrt{3} \approx 4 \times 1.732 \approx 6.928 \text{ cm}
\]

So, the length of the side opposite the 60-degree angle is \( 4\sqrt{3} \) cm or approximately \( 6.93 \) cm.

Thus, the exact answer is:
The side opposite the 60-degree angle is \( 4\sqrt{3} \) cm.