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Zorluk: OrtaTriangles and Pythagorean Theorem

A canvas sail is shaped like a right triangle. The longest side of the sail measures 15 feet15\text{ feet}, and the bottom edge measures 9 feet9\text{ feet}. What is the total area of the canvas sail?

  1. 54 square feet54\text{ square feet}Cevap
  2. B
    67.5 square feet67.5\text{ square feet}
  3. C
    108 square feet108\text{ square feet}
  4. D
    27 square feet27\text{ square feet}

Cevap

54 square feet54\text{ square feet}
The area of a right triangle is half the product of its legs. First, the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) must be used to find the missing perpendicular height (the second leg). The longest side is the hypotenuse (1515), and the bottom edge is a leg (99). Solving for the missing height gives 152−92=225−81=144=12 feet\sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12\text{ feet}. The area is then found by using the formula 12×base×height\frac{1}{2} \times \text{base} \times \text{height}, which equals 12×9×12=54 square feet\frac{1}{2} \times 9 \times 12 = 54\text{ square feet}.

Adım Adım Çözüm

1
Identify the known sides of the right triangle.
The hypotenuse (longest side) is 15 feet15\text{ feet} and the base (one leg) is 9 feet9\text{ feet}.
The Pythagorean theorem requires knowing two sides of a right triangle to find the third.
2
Use the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) to find the missing height (the other leg).
Substituting the known values gives 92+b2=1529^2 + b^2 = 15^2, which simplifies to 81+b2=22581 + b^2 = 225. Solving for b2b^2 gives 144144, meaning the height is 144=12 feet\sqrt{144} = 12\text{ feet}.
The height of the right triangle is necessary to calculate its total area.
3
Calculate the area of the right triangle.
Area = 12×base×height=12×9×12=54 square feet\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 12 = 54\text{ square feet}.
The standard triangle area formula provides the final answer.

Anahtar Kavram

Using the Pythagorean theorem to find a missing side length before calculating the area of a right triangle.
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Konular

Triangles and Pythagorean Theorem

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