Question

if the side opposite a 60-degree angle in a 30-60-90 right triangle is 7 square root of 3 cm long, how long is th hypotenuse?

Answer 1

In a 30-60-90 triangle, the sides are in a specific ratio based on the angles. The side opposite the 30-degree angle (the shortest side) is \(x\), the side opposite the 60-degree angle is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Given that the side opposite the 60-degree angle is \(7\sqrt{3}\) cm, we set this equal to \(x\sqrt{3}\):

\[
x\sqrt{3} = 7\sqrt{3}
\]

To find \(x\), we can divide both sides by \(\sqrt{3}\):

\[
x = 7 \text{ cm}
\]

Now, the hypotenuse is given by \(2x\):

\[
\text{Hypotenuse} = 2x = 2 \times 7 = 14 \text{ cm}
\]

Therefore, the hypotenuse of the triangle is \(14\) cm long.

Answer 2

you sure

Answer 3

Yes, I’m sure. Let's recap the reasoning to confirm:

In a 30-60-90 triangle, the side lengths are in the following ratio:

- The side opposite the 30-degree angle is \(x\).
- The side opposite the 60-degree angle is \(x\sqrt{3}\).
- The hypotenuse is \(2x\).

You stated that the side opposite the 60-degree angle is \(7\sqrt{3}\) cm. Since this corresponds to \(x\sqrt{3}\), we set up the equation:

\[
x\sqrt{3} = 7\sqrt{3}
\]

Dividing both sides by \(\sqrt{3}\) gives us:

\[
x = 7 \text{ cm}
\]

Now, to find the hypotenuse:

\[
\text{Hypotenuse} = 2x = 2 \times 7 = 14 \text{ cm}
\]

Thus, the hypotenuse is indeed \(14\) cm long. If you have further questions or need clarification, feel free to ask!