Question

a pole 9 feet tall is used to support a guy wire for a tower, which runs from the tower to a metal stake in the ground. after placing the pole, corey measures the distance from the pole to the stake and from the pole to the tower, as shown in the diagram below. find the length of the guy wire, to the nearest foot. (diagram is not to scale.)

Explanation:

Step1: Use the concept of similar triangles

Since the two triangles (the small one with height \(9\) ft and base \(9\) ft and the large one with height \(h\) (height of the tower) and base \(9 + 18=27\) ft) are similar. For similar triangles, the ratios of corresponding sides are equal. The ratio of the bases of the two similar triangles is \(\frac{9 + 18}{9}=\frac{27}{9} = 3\). So the height of the tower \(h=3\times9 = 27\) ft.

Step2: Apply the Pythagorean theorem

Let the length of the guy - wire be \(l\). The base of the large right - triangle (formed by the tower, the ground, and the guy - wire) is \(18+9 = 27\) ft and the height is \(27\) ft. By the Pythagorean theorem \(l=\sqrt{27^{2}+27^{2}}\).

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Answer:

\(38\) feet