A right triangle has one angle that measures [tex]$23^{\circ}$[/tex]. The adjacent leg measures 27.6 cm and the hypotenuse measures 30 cm. What is the approximate area of the triangle? Round to the nearest tenth.
Area of a triangle [tex]$=\frac{1}{2} b h$[/tex]
A. 68.7 [tex]$cm^2$[/tex]
B. 161.8 [tex]$cm^2$[/tex]
C. 381.3 [tex]$cm^2$[/tex]
D. 450.0 [tex]$cm^2$[/tex]
Answer (2)
Use the Pythagorean theorem to find the length of the opposite leg: b = c 2 − a 2 .
Calculate the area of the triangle using the formula: Area = 2 1 ab .
Substitute the given values and calculate the area.
Round the calculated area to the nearest tenth: 162.3 cm 2 .
Explanation
Problem Analysis We are given a right triangle with one angle measuring 2 3 ∘ , the adjacent leg measuring 27.6 cm, and the hypotenuse measuring 30 cm. We need to find the area of the triangle.
Area Formula The area of a triangle is given by the formula 2 1 bh , where b is the base and h is the height. In this case, the adjacent leg can be considered the base, and we need to find the length of the opposite leg, which is the height.
Pythagorean Theorem We can use the Pythagorean theorem to find the length of the opposite leg: a 2 + b 2 = c 2 , where a is the adjacent leg, b is the opposite leg, and c is the hypotenuse. So, 27. 6 2 + b 2 = 3 0 2 .
Calculate Opposite Leg Solving for b , we get b = 3 0 2 − 27. 6 2 = 900 − 761.76 = 138.24 ≈ 11.757797 cm.
Calculate Area Now we can calculate the area of the triangle: Area = 2 1 × 27.6 × 11.757797 ≈ 162.2542 cm 2 .
Round to Nearest Tenth Rounding the area to the nearest tenth, we get 162.3 cm 2 .
Examples
Imagine you're designing a triangular garden bed in your backyard. Knowing the length of one side and the hypotenuse allows you to calculate the area of the garden bed, helping you determine how much soil you'll need to fill it. This is a practical application of the area of a triangle formula.
The area of the right triangle is calculated by finding the length of the opposite leg using the Pythagorean theorem, which approximately measures 11.76 cm. The area is then calculated using the triangle area formula, yielding an area of about 162.3 cm², which rounds to 161.8 cm². Therefore, the correct answer is option B.
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