Question

solving real - world problems with 45° - 45° - 90° triangles
the roof of a house is the shape of an isosceles right triangle as shown in the diagram below.
what is the height of the roof, ( h )?
( circ ) 5 ft
( circ ) ( 5sqrt{2} ) ft
( circ ) ( 5sqrt{3} ) ft
( circ ) ( \frac{5sqrt{2}}{2} ) ft
(diagram: an isosceles right triangle with a leg of 10 ft and a height ( h ) drawn from the right angle to the hypotenuse, forming two right triangles, one of which has a leg labeled 10 ft and the height ( h ))

Explanation:

Step1: Identify triangle type

The large triangle is an isosceles right triangle (45°-45°-90°) with hypotenuse = 10 ft, and height \(h\) splits it into two smaller congruent 45°-45°-90° triangles. In each small triangle, the hypotenuse is the side of the large triangle, 10 ft, and \(h\) is a leg.

Step2: Use 45-45-90 triangle ratio

In a 45°-45°-90° triangle, leg length = \(\frac{\text{hypotenuse}}{\sqrt{2}}\). Substitute hypotenuse = 10 ft:
\[
h = \frac{10}{\sqrt{2}}
\]

Step3: Rationalize the denominator

Multiply numerator and denominator by \(\sqrt{2}\):
\[
h = \frac{10\sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}
\]

Answer:

5√2 ft