A right-angle triangle has one angle of 36° and a hypotenuse of 48.2 m. What is the area of the triangle?
a. 497.2 m²
c. 552.4 m²
b. 544.2 m²
d. 553.6 m²
To find the area of a right-angled triangle, we need to know the length of the two perpendicular sides.
Let's label the sides of the triangle as follows:
Hypotenuse = 48.2 m
Angle A = 36°
Side opposite angle A = a (unknown)
Side adjacent to angle A = b (unknown)
In a right-angled triangle, we can use trigonometric ratios to find the lengths of the sides. The sine ratio is useful in this case:
sin(A) = Opposite/Hypotenuse
sin(36°) = a/48.2
Rearranging the equation, we get:
a = sin(36°) * 48.2
a ≈ 0.5878 * 48.2
a ≈ 28.34 m
Now that we know the length of side a, we can use the formula to find the area of a right-angled triangle:
Area = (1/2) * base * height
Area = (1/2) * a * b
To find side b, we can use the Pythagorean theorem:
a² + b² = c²
28.34² + b² = 48.2²
b² ≈ 48.2² - 28.34²
b² ≈ 1654 - 802.2756
b² ≈ 851.7244
b ≈ √851.7244
b ≈ 29.2 m
Now we can calculate the area:
Area = (1/2) * 28.34 * 29.2
Area ≈ 413.748 ≈ 413.7 m²
Therefore, the closest option is a. 497.2 m².