To calculate the perimeter and area of a right-angled triangle with a base of 5 inches and a 59-degree angle, use trigonometry to find the height, then apply the area formula. The area is approximately 10.73 in², and the perimeter is approximately 15.88 inches.
To find the perimeter and area of the right-angled triangle with a base of 5 inches and an angle of 59 degrees opposite to the base, we can use trigonometry and the area formula for triangles. The area (A) of a triangle can be calculated using the formula A = 1/2 × base × height. To find the height, we can use the sine function (since we know an angle and the base) which gives us:
height = base × sin(angle).
Height = 5 in × sin(59°)
= 5 in × 0.8572 (using a calculator)
= 4.29 inches (rounded to two decimal places).
Now that we have the height, we can calculate the area:
Area = 1/2 × base × height
= 1/2 × 5 in × 4.29 in
= 10.725 in², which, when rounded to two decimal places, is 10.73 in².
To find the perimeter, we need to calculate the length of the hypotenuse using the Pythagorean theorem. Let's call the hypotenuse 'c', then we have
c = √(base² + height²)
= √(5² + 4.29²)
= √(25 + 18.4041)
= √(43.4041)
= 6.59 in (rounded to two decimal places). Now, the perimeter is the sum of all sides:
Perimeter = base + height + hypotenuse
= 5 in + 4.29 in + 6.59 in
= 15.88 in, rounded to two decimal places gives us 15.88 in.