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Zorluk: OrtaTriangles and Pythagorean Theorem

A painter leans a 25-foot25\text{-foot} ladder against a vertical wall. The base of the ladder is placed exactly 7 feet7\text{ feet} away from the base of the wall on level ground. How high up the wall, in feet, does the top of the ladder reach?

Cevap: 24 feet

Cevap

The top of the ladder reaches 24 feet up the wall.
The correct answer is derived by modeling the scenario as a right triangle. The ladder acts as the hypotenuse (c=25c = 25) and the distance along the ground acts as one leg (a=7a = 7). Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we solve for the missing height (bb). Substituting the values gives 72+b2=2527^2 + b^2 = 25^2, which simplifies to 49+b2=62549 + b^2 = 625. Subtracting 49 yields b2=576b^2 = 576. The square root of 576 is 24, meaning the ladder reaches 24 feet up the wall.

Adım Adım Çözüm

1
Set up the right triangle relationship.
a2+b2=c2a^2 + b^2 = c^2, where c=25c = 25 and a=7a = 7
The wall is perpendicular to the ground, forming a right triangle with the ladder acting as the hypotenuse.
2
Substitute known values into the Pythagorean theorem.
72+b2=2527^2 + b^2 = 25^2
We know the hypotenuse (2525) and one leg (77), and we need to find the other leg (bb).
3
Square the known numbers.
49+b2=62549 + b^2 = 625
Evaluating the exponents is required before isolating the variable term.
4
Isolate the variable term.
b2=625−49=576b^2 = 625 - 49 = 576
Subtracting 4949 from both sides leaves b2b^2 by itself.
5
Solve for the height.
b=576=24b = \sqrt{576} = 24
Taking the square root of both sides gives the length of the missing side.

Anahtar Kavram

Using the Pythagorean theorem to find a missing leg of a right triangle in a real-world scenario.
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Konular

Triangles and Pythagorean Theorem

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