Question

which triangle is a 30° - 60° - 90° triangle?

Explanation:

Step1: Recall side - ratio property

In a 30° - 60° - 90° triangle, if the shorter leg (opposite the 30° angle) has length $a$, the longer leg (opposite the 60° angle) has length $a\sqrt{3}$, and the hypotenuse (opposite the 90° angle) has length $2a$.

Step2: Check each option

For the first option: If $a = 5$, then $a\sqrt{3}=5\sqrt{3}$ and $2a = 10$. The side - lengths 5, $5\sqrt{3}$, 10 satisfy the ratio of a 30° - 60° - 90° triangle.
For the second option: If $a = 5$, then $2a=10
eq15$, so it's not a 30° - 60° - 90° triangle.
For the third option: If $a = 5$, then $a\sqrt{3}=5\sqrt{3}
eq10\sqrt{3}$, so it's not a 30° - 60° - 90° triangle.
For the fourth option: If $a = 10$, then $a\sqrt{3}=10\sqrt{3}
eq5\sqrt{3}$, so it's not a 30° - 60° - 90° triangle.

Answer:

The triangle with side - lengths 5, $5\sqrt{3}$, 10 is a 30° - 60° - 90° triangle.