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Difficulty: MediumRight Triangles and the Pythagorean Theorem

A 2525-foot ladder is leaning against a vertical wall such that the base of the ladder is 77 feet away from the base of the wall. If the top of the ladder slides down the wall by 44 feet, how many feet does the base of the ladder slide outward along the ground?

  1. A
    44
  2. 88Answer
  3. C
    99
  4. D
    1515

Answer

The base of the ladder slides outward by 88 feet.
The correct answer is 88 feet. By applying the Pythagorean theorem, the initial height of the ladder is found to be 252−72=24\sqrt{25^2 - 7^2} = 24 feet. When the ladder slides down 44 feet, its new height is 2020 feet. Using the Pythagorean theorem again, the new base distance is 252−202=15\sqrt{25^2 - 20^2} = 15 feet. The difference between the new base distance and the initial base distance is 15−7=815 - 7 = 8 feet.

Step-by-Step Solution

1
Find the initial height of the top of the ladder on the wall using the Pythagorean theorem.
Initial Height=252−72=625−49=576=24\text{Initial Height} = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24 feet
The wall, the ground, and the ladder form a right triangle where the ladder is the hypotenuse (2525 feet) and the base distance is one leg (77 feet).
2
Calculate the new height of the top of the ladder after it slides down.
New Height=24−4=20\text{New Height} = 24 - 4 = 20 feet
The top of the ladder slides down by 44 feet from its initial height of 2424 feet.
3
Calculate the new distance of the base of the ladder from the wall using the Pythagorean theorem.
New Base Distance=252−202=625−400=225=15\text{New Base Distance} = \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = \sqrt{225} = 15 feet
The ladder length (hypotenuse) remains 2525 feet, and the new height (one leg) is 2020 feet.
4
Find the distance the base of the ladder slid outward by subtracting the initial base distance from the new base distance.
Outward Slide=15−7=8\text{Outward Slide} = 15 - 7 = 8 feet
The base was initially 77 feet from the wall and is now 1515 feet from the wall.

Key Concept

Applying the Pythagorean theorem to relate side lengths in changing right triangle configurations.
Estimated Time:1m 30s
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Topics

Right Triangles and the Pythagorean Theorem

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