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Difficulty: MediumRight Triangle Trigonometry (SOHCAHTOA)

A diagonal support beam is installed to stabilize a wooden wall frame. The beam extends from the top-left corner of the frame to the bottom-right corner, forming a right triangle with the top horizontal beam and the right vertical post. The vertical post has a height of 1515 feet. If the angle θ\theta between the diagonal support beam and the top horizontal beam satisfies tan⁡(θ)=34\tan(\theta) = \frac{3}{4}, what is the length, in feet, of the diagonal support beam?

  1. A
    11.2511.25
  2. B
    2020
  3. 2525Answer
  4. D
    3535
  5. E
    18.7518.75

Answer

The length of the diagonal support beam is 2525 feet.
In the right triangle formed by the wall frame and diagonal beam, angle θ\theta is between the diagonal beam (hypotenuse) and the top horizontal beam (adjacent side). The vertical post is opposite to angle θ\theta, with a length of 1515 feet. Using tan⁡(θ)=OppositeAdjacent=34\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4}, we substitute 1515 for the opposite side to get 34=15Adjacent\frac{3}{4} = \frac{15}{\text{Adjacent}}, which yields an adjacent side length of 2020 feet. Using the Pythagorean theorem, the hypotenuse is 152+202=225+400=625=25\sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25 feet.

Step-by-Step Solution

1
Identify the given trigonometric relationship and right triangle sides.
The vertical post is opposite to angle θ\theta, so Opposite=15\text{Opposite} = 15 feet. The top horizontal beam is adjacent to θ\theta, and the diagonal support beam is the hypotenuse.
Understanding side positions relative to angle θ\theta is essential for setting up the tangent ratio correctly.
2
Use the definition of tangent to find the length of the adjacent side (horizontal beam).
tan⁡(θ)=OppositeAdjacent  ⟹  34=15Adjacent  ⟹  Adjacent=15×43=20\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \implies \frac{3}{4} = \frac{15}{\text{Adjacent}} \implies \text{Adjacent} = \frac{15 \times 4}{3} = 20 feet.
Solving for the adjacent side gives the horizontal leg of the right triangle.
3
Apply the Pythagorean theorem to calculate the length of the diagonal support beam (hypotenuse).
Hypotenuse=152+202=225+400=625=25\text{Hypotenuse} = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25 feet.
The diagonal beam forms the hypotenuse of the right triangle with legs of length 1515 feet and 2020 feet.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA) and the Pythagorean Theorem
Estimated Time:1m 15s
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Topics

Right Triangle Trigonometry (SOHCAHTOA)

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