Question

use special right triangle ratios to find the length of the hypotenuse.
(image of a right triangle with a right angle, one angle 45°, the other angle 45°, and the leg adjacent to the right angle and 45° angle labeled 11)
a. $11\sqrt{3}$
b. 22
c. 11
d. $11\sqrt{2}$

Explanation:

Step1: Identify Triangle Type

The triangle has two \(45^\circ\) angles and a right angle, so it's a \(45^\circ - 45^\circ - 90^\circ\) (isosceles right) triangle.

Step2: Recall Special Ratio

In a \(45^\circ - 45^\circ - 90^\circ\) triangle, the ratio of legs to hypotenuse is \(1:1:\sqrt{2}\) (leg : leg : hypotenuse).

Step3: Apply Ratio to Given Leg

One leg is \(11\). Let hypotenuse be \(h\). By ratio, \(h=\text{leg}\times\sqrt{2}=11\sqrt{2}\).

Answer:

D. \(11\sqrt{2}\)