The hypotenuse if a right triangle measures 9 inches and one of the acute angles measures 36 degress. What is the area if the triangle? Round to the nearest square inch.
I'll find one side.
Triangle ABC,
angle A = 36
side c (hypotenuse) = 9
To find side a (opposite side)
sin A = op/hyp = a/c = a/9
sin 36 = a/9
0.5878 = a/9
9 * 0.5878 = a
a = 5.2902
Area = 1/2 bh
In this case, (right triangle)
A = 1/2 ab (side a, side b)
I found side a, you need to find side b.
To find the area of a right triangle, we need to know the lengths of the two legs or the lengths of one leg and the hypotenuse. In this case, we know the length of the hypotenuse (9 inches) and one of the acute angles (36 degrees). However, we do not have the lengths of the legs directly.
To solve this problem, we can use trigonometric ratios, such as sine, cosine, or tangent. In a right triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the hypotenuse.
Let's label the sides of the triangle. Let "x" represent the length of one leg and "y" represent the length of the other leg. The side opposite the 36-degree angle is "x" and the hypotenuse is 9 inches.
Using the sine function, we can write:
sin(36 degrees) = x / 9
To find "x," we can rearrange the equation:
x = 9 * sin(36 degrees)
Now, substitute the given values into the equation:
x = 9 * sin(36 degrees)
x = 9 * 0.5878 (using a calculator to find the sine of 36 degrees, rounded to four decimal places)
x ≈ 5.2902
So, one leg of the right triangle measures approximately 5.2902 inches.
To find the other leg, we can use the Pythagorean theorem, which states that the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.
Using the Pythagorean theorem:
x^2 + y^2 = 9^2
(5.2902)^2 + y^2 = 81
27.9936 + y^2 = 81
y^2 = 81 - 27.9936
y^2 ≈ 53.0064
To find "y," we can take the square root of both sides of the equation:
y ≈ √(53.0064)
y ≈ 7.2801
So, the other leg of the right triangle measures approximately 7.2801 inches.
Now that we know the lengths of the two legs, we can finally calculate the area of the triangle using the formula:
Area = (base * height) / 2
In this case, we can choose any leg as the base and the other leg as the height. Let's choose the leg measuring 5.2902 inches as the base and the leg measuring 7.2801 inches as the height.
Area ≈ (5.2902 * 7.2801) / 2
Area ≈ 38.4914 / 2
Area ≈ 19.2457
Rounding to the nearest square inch, the area of the triangle is approximately 19 square inches.