Question

If the side opposite a 60-degree angle in a 30-60-90 right triangle is 73–√ cm
long, how long is the hypotenuse?(1 point)
Responses

28 cm
28 cm

14 cm
14 cm

73–√ cm
7 Start Root 3 End Root cm

7 cm

Answer 1

In a 30-60-90 right triangle, the ratios of the lengths of the sides are as follows:

- The side opposite the 30-degree angle (the shortest side) is \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse (opposite the 90-degree angle) is \( 2x \).

In this case, the side opposite the 60-degree angle is given as \( 73 - \sqrt{cm} \). According to the ratio, we can set up the equation:

\[
x\sqrt{3} = 73 - \sqrt{cm}
\]

To find \( x \), we can rearrange this into:

\[
x = \frac{73 - \sqrt{cm}}{\sqrt{3}}
\]

Then, we can find the hypotenuse \( 2x \):

\[
\text{Hypotenuse} = 2x = 2 \left(\frac{73 - \sqrt{cm}}{\sqrt{3}}\right) = \frac{2(73 - \sqrt{cm})}{\sqrt{3}} = \frac{146 - 2\sqrt{cm}}{\sqrt{3}}
\]

Since we are only given options, we cannot determine the exact numerical value unless \( \sqrt{cm} \) is specified. However, the hypotenuse can be calculated based on the derived expression above.

If we assume \( 73 - \sqrt{cm} \) equals \( 14\sqrt{3} \) based on common measurements in these triangles, we can set \( x = 14 \):

\[
\text{Hypotenuse} = 2 \cdot 14 = 28
\]

Thus, the hypotenuse in your case is:

**28 cm**.