A pole 10 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, Suav 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.

The problem has been modeled as 2 similar right triangles. The smaller right triangle has a base of 3 ft and a height of 10 ft. The larger right triangle has a base of 12 ft.

Pythagoras Theorem: [tex]a^2+b^2=c^2[/tex]

(where a and b are the legs, and c is the hypotenuse, of a right triangle)

Use Pythagoras Theorem to calculate the hypotenuse of the smaller triangle.

Given:

a = 3 ft
b = 9 ft

Substitute the given values into the formula and solve for the hypotenuse (c):

[tex]\implies \sf 3^2+10^2=c^2[/tex]

[tex]\implies \sf c^2=109[/tex]

[tex]\implies \sf c=\sqrt{109}\:\:ft[/tex]

Similar Triangle Theorem

If two triangles are similar, the ratio of their corresponding sides is equal.

[tex]\implies \sf hypotenuse_{large}:base_{large}=hypotenuse_{small}:base_{small}[/tex]

[tex]\implies \sf c:12=\sqrt{109}:3[/tex]

[tex]\implies \sf \dfrac{c}{12}=\dfrac{\sqrt{109}}{3}[/tex]

[tex]\implies \sf c=\dfrac{12\sqrt{109}}{3}[/tex]

[tex]\implies \sf c=42\:ft\:\:(nearest\:foot)[/tex]

Therefore, the length of the guy wire (hypotenuse of the largest right triangle) to the nearest foot it 42 feet.

Learn more about similar triangles here:

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