Question

when constructing a building, rafters are used to support the roof. the rafter shown in the diagram has the shape of an isosceles triangle. what is the height of the roof? (squaresquare ft)

Explanation:

Step1: Find the base length of the right - triangle

Since the triangle \(NRQ\) is isosceles and \(NP\perp RQ\), \(RP=\frac{RQ}{2}\). Given \(RQ = 60\) ft, then \(RP=\frac{60}{2}=30\) ft.

Step2: Apply the Pythagorean theorem

In right - triangle \(NPQ\), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = NQ=61\) ft (hypotenuse) and \(a = PQ = 30\) ft. Let \(h=NP\) (height). Then \(h=\sqrt{NQ^{2}-PQ^{2}}\).
Substitute \(NQ = 61\) and \(PQ = 30\) into the formula: \(h=\sqrt{61^{2}-30^{2}}=\sqrt{(61 + 30)(61 - 30)}=\sqrt{91\times31}=\sqrt{59\times59}=59\) (using \(a^{2}-b^{2}=(a + b)(a - b)\) and \(61^{2}-30^{2}=3721-900 = 2821\), \(\sqrt{2821}=59\)).

Answer:

\(59\)