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Difficulty: Very hardTriangles and Pythagorean Theorem

Solve the following word problem by applying properties of right triangles.

Answer:A 25-foot ladder is leaning against a vertical wall, with its base placed 7 feet away from the bottom of the wall. If the top of the ladder slides exactly 4 feet straight down the wall, the base of the ladder will slide 【8】 feet further away from the wall.

Answer

8
By applying the Pythagorean theorem to the initial position, we establish that the ladder reached a height of 24 feet (252−72=24225^2 - 7^2 = 24^2). When it slips 4 feet down, the new height becomes 20 feet. Using the Pythagorean theorem again for the new position with a constant hypotenuse of 25, the new base distance is 15 feet (252−202=15225^2 - 20^2 = 15^2). The ladder started 7 feet away, so it moved an additional 8 feet (15−7=815 - 7 = 8).

Step-by-Step Solution

1
Find the initial height of the top of the ladder on the wall using the Pythagorean theorem.
a2+72=252a^2 + 7^2 = 25^2, which simplifies to a2+49=625a^2 + 49 = 625. Solving for a2a^2 gives 576576, so a=24a = 24 feet.
The ladder, the wall, and the ground form a right triangle where the ladder is the hypotenuse.
2
Calculate the new height of the top of the ladder after it slides down the wall.
24−4=2024 - 4 = 20 feet.
The top of the ladder slid exactly 4 feet downwards, reducing the vertical leg of our right triangle.
3
Find the new distance of the base of the ladder from the wall.
b2+202=252b^2 + 20^2 = 25^2, which simplifies to b2+400=625b^2 + 400 = 625. Solving for b2b^2 gives 225225, so b=15b = 15 feet.
The ladder's overall length does not change, creating a new right triangle with a height of 20 and a hypotenuse of 25.
4
Calculate the difference between the new base distance and the original base distance.
15−7=815 - 7 = 8 feet.
The question specifically asks how much further the base slid away from the wall, not the final total distance.

Key Concept

Applying the Pythagorean theorem iteratively to linked right triangles that share a common hypotenuse length
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Topics

Triangles and Pythagorean Theorem

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