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Difficulty: EasyTriangles and Pythagorean Theorem

A guy wire is attached to the top of a 1212-foot vertical pole. The wire is anchored to the ground 55 feet from the base of the pole. What is the length of the guy wire?

  1. 1313 feetAnswer
  2. B
    119\sqrt{119} feet
  3. C
    1717 feet
  4. D
    169169 feet

Answer

The guy wire is 1313 feet long.
The length of the guy wire represents the hypotenuse of a right triangle formed by the pole and the ground. According to the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), the square of the hypotenuse equals the sum of the squares of the two legs. Substituting the given values: 122+52=144+25=16912^2 + 5^2 = 144 + 25 = 169. Taking the square root of 169169 gives 1313 feet.

Step-by-Step Solution

1
Identify the known values representing the legs of the right triangle.
Leg a=5a = 5 feet, Leg b=12b = 12 feet.
The pole and the ground meet at a right angle, so they form the legs of the triangle.
2
Apply the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2).
52+122=c25^2 + 12^2 = c^2
The theorem connects the lengths of the legs to the hypotenuse (the guy wire).
3
Square the lengths of the legs and sum them.
25+144=16925 + 144 = 169
This calculates the value of the hypotenuse squared (c2c^2).
4
Take the square root of the sum.
c=169=13c = \sqrt{169} = 13
This isolates cc and provides the actual length of the hypotenuse.

Key Concept

The Pythagorean theorem is used to find the hypotenuse of a right triangle when both legs are known.
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Topics

Triangles and Pythagorean Theorem

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