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Zorluk: KolayPythagorean Theorem and Special Right Triangles

A 1313-foot ladder is leaning against a flat vertical wall. The base of the ladder is placed 55 feet away from the bottom of the wall. How many feet up the wall does the ladder reach?

  1. A
    8\sqrt{8}
  2. B
    88
  3. 1212Cevap
  4. D
    194\sqrt{194}
  5. E
    1818

Cevap

The height the ladder reaches is 1212 feet.
The correct answer is 1212 feet. The ladder, wall, and ground form a right triangle where the ladder is the hypotenuse (1313 feet) and the distance along the ground is one leg (55 feet). By the Pythagorean theorem, the height up the wall, bb, satisfies 52+b2=1325^2 + b^2 = 13^2. Solving for bb gives b2=169−25=144b^2 = 169 - 25 = 144, so b=12b = 12.

Adım Adım Çözüm

1
Identify the hypotenuse and the given leg from the word problem description.
The ladder length is the hypotenuse (c=13c = 13), and the distance from the wall is one of the legs (a=5a = 5).
The ladder forms the diagonal side opposite the right angle formed by the vertical wall and the ground.
2
Set up the Pythagorean theorem to find the unknown leg length.
52+b2=1325^2 + b^2 = 13^2.
The Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) relates the side lengths of any right triangle.
3
Solve for the unknown leg length bb by simplifying terms and taking the square root.
25+b2=169  ⟹  b2=144  ⟹  b=144=1225 + b^2 = 169 \implies b^2 = 144 \implies b = \sqrt{144} = 12.
Subtracting the square of the known leg from the square of the hypotenuse isolates the squared unknown leg, which can then be solved by finding its square root.

Anahtar Kavram

Using the Pythagorean theorem to find an unknown leg of a right triangle when the hypotenuse and one leg are known.

Alternatif Yöntem

Recognizing that 55 and 1313 are part of the common Pythagorean triple 55-1212-1313 allows you to immediately identify the missing leg as 1212 without performing calculations.
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Pythagorean Theorem and Special Right Triangles

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