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√2 ft 5√3 ft 5√2/2 ft

Explanation:

Step1: Analyze the isosceles right - triangle properties

In an isosceles right - triangle, the height \(h\) also acts as the median. The hypotenuse of the large isosceles right - triangle (the base of the roof) can be found using the Pythagorean theorem. But we can also use the fact that the height \(h\) of an isosceles right - triangle (divided by the altitude) forms two smaller isosceles right - triangles.
For the smaller right - triangle (half of the roof triangle), the hypotenuse is \(10\) ft. Let the legs of the smaller isosceles right - triangle be \(h\) (height) and the base segment. In an isosceles right - triangle with hypotenuse \(c\) and leg \(a\), the formula is \(c=\sqrt{a^{2}+a^{2}}=\sqrt{2a^{2}} = a\sqrt{2}\).

Step2: Solve for \(h\)

If \(c = 10\) (hypotenuse of the smaller isosceles right - triangle) and \(c=a\sqrt{2}\) (where \(a = h\)), then \(10=h\sqrt{2}\). Solving for \(h\), we get \(h=\frac{10}{\sqrt{2}}\). Rationalizing the denominator: \(h=\frac{10\sqrt{2}}{2}=5\sqrt{2}\) ft.

Answer:

\(5\sqrt{2}\text{ ft}\) (the second option)