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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A vertical flagpole is secured by two straight guide wires anchored to the flat ground on opposite sides of the pole. The first guide wire is 1313 feet long and its anchor is 55 feet from the base of the pole. The second guide wire is anchored such that it makes a 30∘30^\circ angle of elevation with the ground. If both guide wires are attached to the flagpole at the same height, what is the length, in feet, of the second guide wire?

Answer: 24 feet

Answer

The length of the second guide wire is 2424 feet.
The correct answer is 2424. First, the height of the attachment point is found using the Pythagorean theorem: h=132−52=12h = \sqrt{13^2 - 5^2} = 12 feet. Since the second wire makes a 30∘30^\circ angle of elevation with the ground, it forms a 30-60-90 right triangle where the flagpole height of 1212 feet is the leg opposite the 30∘30^\circ angle. The length of the wire is the hypotenuse of this triangle, which is twice the length of the opposite leg: 2×12=242 \times 12 = 24 feet.

Step-by-Step Solution

1
Use the Pythagorean theorem to calculate the height of the flagpole where the guide wires are attached.
The flagpole height is 1212 feet.
The first guide wire, the flagpole, and the ground form a right triangle with a hypotenuse of 1313 feet and a horizontal leg of 55 feet. Thus, h=132−52=169−25=12h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = 12.
2
Apply the properties of a 30-60-90 special right triangle to find the length of the second guide wire.
The length of the second guide wire is 2424 feet.
The second wire forms a 30-60-90 right triangle with the flagpole and the ground. The angle of elevation is 30∘30^\circ, which means the side opposite this angle is the vertical height of the flagpole (1212 feet). In a 30-60-90 triangle, the hypotenuse (the wire length) is twice the length of the shorter leg (opposite the 30∘30^\circ angle), so the length is 2×12=242 \times 12 = 24.

Key Concept

Applying the Pythagorean Theorem and the ratio properties of 30-60-90 special right triangles to solve multi-step geometry problems.
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Topics

Pythagorean Theorem and Special Right Triangles

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