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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A rectangular billboard frame ABCDABCD has a length AB=16AB = 16 feet and width BC=12BC = 12 feet. A straight diagonal support brace connects vertex AA to vertex CC. To add structural stability, a secondary beam is installed perpendicular to diagonal ACAC, extending from vertex BB to meet ACAC at point PP. What is the length, in feet, of segment BPBP?

  1. 9.6Answer
  2. B
    10.0
  3. C
    14.0
  4. D
    19.2
  5. E
    20.0

Answer

9.6 feet
The hypotenuse ACAC of right triangle ABCABC equals 162+122=20\sqrt{16^2 + 12^2} = 20 feet. Since the area of triangle ABCABC can be calculated either as 12×16×12=96\frac{1}{2} \times 16 \times 12 = 96 or as 12×20×BP\frac{1}{2} \times 20 \times BP, solving 10×BP=9610 \times BP = 96 gives BP=9.6BP = 9.6 feet.

Step-by-Step Solution

1
Calculate the length of diagonal ACAC using the Pythagorean theorem.
AC=AB2+BC2=162+122=256+144=400=20AC = \sqrt{AB^2 + BC^2} = \sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20 feet.
Triangle ABCABC is a right triangle with right angle at BB and hypotenuse ACAC.
2
Express the area of triangle ABCABC using the two legs.
Area=12×AB×BC=12×16×12=96\text{Area} = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 16 \times 12 = 96 square feet.
The area of a right triangle is half the product of its perpendicular legs.
3
Express the area using hypotenuse ACAC as the base and BPBP as the height, then solve for BPBP.
Area=12×AC×BP  ⟹  96=12×20×BP  ⟹  10×BP=96  ⟹  BP=9.6\text{Area} = \frac{1}{2} \times AC \times BP \implies 96 = \frac{1}{2} \times 20 \times BP \implies 10 \times BP = 96 \implies BP = 9.6 feet.
Segment BPBP is given as perpendicular to base ACAC.

Key Concept

Altitude to the Hypotenuse in a Right Triangle
Estimated Time:1m 15s
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Topics

Pythagorean Theorem and Special Right Triangles

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