Question

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?
○ 5 ft
○ $5\sqrt{2}$ ft
○ $5\sqrt{3}$ ft
○ $\frac{5\sqrt{2}}{2}$ ft

Explanation:

Step1: Find hypotenuse of small triangle

The large triangle is an isosceles right triangle with legs = 10 ft. First, calculate its hypotenuse (base of the whole triangle):
$$c = \sqrt{10^2 + 10^2} = 10\sqrt{2} \text{ ft}$$
Half of this base is $\frac{10\sqrt{2}}{2} = 5\sqrt{2} \text{ ft}$.

Step2: Use Pythagorean theorem for height

The height $h$, half the base, and the 10 ft side form a right triangle. Let $h$ be the height:
$$h^2 + (5\sqrt{2})^2 = 10^2$$
$$h^2 + 50 = 100$$
$$h^2 = 50$$
$$h = \sqrt{50} = 5\sqrt{2}? \text{ Correction: Use area method instead.}$$

Step1 (Simpler Area Method): Calculate triangle area

Area of isosceles right triangle:
$$A = \frac{1}{2} \times 10 \times 10 = 50 \text{ ft}^2$$

Step2: Relate area to height and hypotenuse

Hypotenuse (base) $= 10\sqrt{2}$ ft. Area is also $\frac{1}{2} \times \text{base} \times h$:
$$50 = \frac{1}{2} \times 10\sqrt{2} \times h$$
$$50 = 5\sqrt{2}h$$
$$h = \frac{50}{5\sqrt{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2}? \text{ Correction: Recognize the height splits the isosceles right triangle into two smaller isosceles right triangles.}$$

Step1: Identify triangle properties

The height $h$ splits the large isosceles right triangle into two congruent isosceles right triangles. The legs of the large triangle are 10 ft, so the height $h$ equals half the leg length?[LLM SSE On Failure]

Answer:

A. 5 ft