Find the perimeter of a right triangle which has a side 7 in. long and the angle opposite to it measures 37 degrees.
To find the perimeter of a right triangle, we need to know the lengths of all three sides. The given information tells us that one side is 7 inches long, but we still need the lengths of the other two sides.
In a right triangle, one of the angles measures 90 degrees. Since the angle opposite the side of length 7 inches is 37 degrees, the remaining angle must be 90 - 37 = 53 degrees.
To find the lengths of the other two sides, we can use trigonometric ratios. In this case, we can use the sine ratio.
The sine ratio in a right triangle is defined as the ratio of the length of the side opposite an angle to the length of the hypotenuse.
In our triangle, the side opposite the angle of 37 degrees is 7 inches. Let's call the length of the hypotenuse "h" and the length of the side opposite the angle of 53 degrees "a".
Applying the sine ratio, we can write:
sin(37 degrees) = 7 / h
To find h, we can rearrange the equation:
h = 7 / sin(37 degrees)
Now, to find the length of the side opposite the angle of 53 degrees, we can use the sine ratio again:
sin(53 degrees) = a / h
Substituting the value of h we found earlier:
sin(53 degrees) = a / (7 / sin(37 degrees))
Simplifying the equation, we get:
a = (7 * sin(53 degrees)) / sin(37 degrees)
Using a calculator, we can find the values of sin(53 degrees) and sin(37 degrees):
sin(53 degrees) ≈ 0.7986
sin(37 degrees) ≈ 0.6018
Substituting these values into the equation, we get:
a = (7 * 0.7986) / 0.6018
≈ 9.3011
So, the length of the side opposite the angle of 53 degrees is approximately 9.3011 inches.
Now, we have the lengths of all three sides: 7 inches, 9.3011 inches, and the hypotenuse, which we found to be 7 / sin(37 degrees) inches.
The perimeter of a triangle is the sum of the lengths of all its sides. Therefore, the perimeter of this right triangle is:
Perimeter = 7 + 9.3011 + 7 / sin(37 degrees)
Using a calculator, we can find the value of sin(37 degrees):
sin(37 degrees) ≈ 0.6018
Substituting this value into the equation, we get:
Perimeter ≈ 7 + 9.3011 + 7 / 0.6018
≈ 7 + 9.3011 + 11.625
Calculating further, the perimeter of the right triangle is approximately:
Perimeter ≈ 27.9261 inches
So, the perimeter of the right triangle is approximately 27.9261 inches.