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Zorluk: KolayRight Triangles and the Pythagorean Theorem

A straight ladder is leaning against a vertical wall. The base of the ladder is placed 99 feet from the bottom of the wall. If the top of the ladder touches the wall at a height of 1212 feet above the ground, what is the length, in feet, of the ladder?

Cevap: 15 feet

Cevap

The length of the ladder is 1515 feet.
By representing the scenario as a right triangle, the two perpendicular sides (legs) have lengths of 99 feet and 1212 feet. Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we find 92+122=81+144=2259^2 + 12^2 = 81 + 144 = 225. Taking the square root of 225225 gives the hypotenuse length, which is 1515 feet.

Adım Adım Çözüm

1
Model the scenario using a right triangle.
A right triangle where the vertical leg is 1212 feet (height on the wall), the horizontal leg is 99 feet (distance along the ground), and the hypotenuse is cc (the ladder length).
The intersection of the vertical wall and the horizontal ground forms a right angle (90∘90^\circ).
2
Set up the Pythagorean equation.
92+122=c29^2 + 12^2 = c^2
The Pythagorean theorem states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.
3
Solve for the hypotenuse cc.
c=81+144=225=15c = \sqrt{81 + 144} = \sqrt{225} = 15
Simplifying the arithmetic yields 225225, and taking the square root gives the final length of 1515 feet.

Anahtar Kavram

Pythagorean Theorem
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Right Triangles and the Pythagorean Theorem

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