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? ft

hint: nr ≅ nq

Explanation:

Step1: Find the base length of the right - triangle

Since the triangle \(NRQ\) is isosceles and \(NP\perp RQ\), \(RP = PQ\). Given \(RQ=24\) ft, then \(PQ=\frac{24}{2}=12\) ft.

Step2: Apply the Pythagorean theorem

In right - triangle \(NPQ\), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = NQ = 25\) ft and \(a = PQ=12\) ft, and \(b = NP\) (the height we want to find).

$$NP=\sqrt{NQ^{2}-PQ^{2}}$$

$$NP=\sqrt{25^{2}-12^{2}}$$

$$NP=\sqrt{(25 + 12)(25 - 12)}$$

(using \(a^{2}-b^{2}=(a + b)(a - b)\))

$$NP=\sqrt{37\times13}$$

$$NP=\sqrt{625 - 144}$$

$$NP=\sqrt{481}$$

$$NP = 20$$

(since \(25^{2}=625\), \(12^{2}=144\), \(625-144 = 481\), and \(\sqrt{481}\approx20\) as \(20^{2}=400\), \(21^{2}=441\), \(22^{2}=484\))

Answer:

\(20\)