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Zorluk: KolayPythagorean Theorem Word Problems

A builder uses a 15-foot diagonal support beam to brace a vertical wall. If the base of the beam rests on the flat floor exactly 9 feet away from the bottom of the wall, at what height on the wall, in feet, does the top of the beam attach?

Cevap: 12 feet

Cevap

12
The physical scenario forms a right triangle where the hypotenuse (the beam) measures 15 feet and one leg (along the floor) measures 9 feet. By using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we can find the missing leg by solving 92+b2=1529^2 + b^2 = 15^2. This simplifies to 81+b2=22581 + b^2 = 225, which means b2=144b^2 = 144. Taking the square root of 144 gives a wall height of 12 feet.

Adım Adım Çözüm

1
Identify the parts of the right triangle.
Hypotenuse c=15c = 15 feet, base leg a=9a = 9 feet, vertical leg bb is unknown.
The beam forms the hypotenuse opposite the right angle made by the intersection of the vertical wall and flat floor.
2
Apply the Pythagorean theorem.
92+b2=1529^2 + b^2 = 15^2
The Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) relates the lengths of the legs of a right triangle to its hypotenuse.
3
Square the known values.
81+b2=22581 + b^2 = 225
To solve for the unknown leg bb, we must first evaluate the squared terms.
4
Isolate the unknown squared variable.
b2=225−81=144b^2 = 225 - 81 = 144
Subtracting 81 from both sides leaves b2b^2 by itself.
5
Take the square root to find the height.
b=144=12b = \sqrt{144} = 12
Taking the square root of b2b^2 gives the actual length of the wall height.

Anahtar Kavram

Using the Pythagorean theorem to find the missing leg of a right triangle.
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Pythagorean Theorem Word Problems

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