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]$=\frac{1}{2} bh$[/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)
The approximate area of the triangle is calculated to be 161.8 cm². This is done using the sine function to find the length of the opposite leg and then applying the area formula. Therefore, the chosen answer is B. 161.8 cm².
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Define the given values: adjacent leg a = 27.6 cm, hypotenuse c = 30 cm, and angle θ = 2 3 ∘ .
Calculate the length of the opposite leg using the sine function: b = c ⋅ sin ( θ ) = 30 ⋅ sin ( 2 3 ∘ ) ≈ 11.721 cm.
Compute the area of the triangle using the formula: Area = 2 1 ab = 2 1 ( 27.6 ) ( 11.721 ) ≈ 161.7498 c m 2 .
Round the area to the nearest tenth: 161.8 c m 2 .
Explanation
Problem Analysis We are given a right triangle with one angle measuring 2 3 ∘ . The adjacent leg to this angle measures 27.6 cm, and the hypotenuse measures 30 cm. Our goal is to find the approximate area of this triangle, rounded to the nearest tenth. The area of a triangle is given by the formula 2 1 bh , where b is the base and h is the height.
Define Given Values Let's denote the given angle as θ = 2 3 ∘ , the adjacent leg as a = 27.6 cm, and the hypotenuse as c = 30 cm. We need to find the length of the opposite leg, which we'll call b , to calculate the area. Since it's a right triangle, we can use trigonometric ratios.
Find the Opposite Leg We can use the sine function to find the length of the opposite leg: sin ( θ ) = c b . Solving for b , we get b = c ⋅ sin ( θ ) = 30 ⋅ sin ( 2 3 ∘ ) .
Calculate the Value of Opposite Leg Using a calculator, we find that sin ( 2 3 ∘ ) ≈ 0.3907 . Therefore, b ≈ 30 ⋅ 0.3907 ≈ 11.721 .
Calculate the Area Now we can calculate the area of the triangle using the formula: Area = 2 1 ab = 2 1 ( 27.6 ) ( 11.721 ) ≈ 2 1 ( 323.4996 ) ≈ 161.7498 .
Round to Nearest Tenth Rounding the calculated area to the nearest tenth, we get 161.8 c m 2 .
Examples
Understanding the area of triangles is crucial in many real-world applications. For example, architects use it to calculate the surface area of triangular structures in building designs. Similarly, surveyors use it to determine the area of land plots that are triangular. In construction, knowing the area helps in estimating the amount of material needed for roofing or paving triangular sections. This concept is also vital in fields like navigation and cartography, where calculating triangular areas aids in mapping and determining distances.
