Find the perimeter and the area of the figure.
A right-angled trapezoid has a shorter base of 7 centimeters, longer base of 9.5 centimeters, and height of 6 centimeters. The trapezoid is divided into a rectangle and a right triangle. The height of the triangle is aligned with the width of the rectangle and it has a hypotenuse of 6.5 centimeters.
perimeter:
cm
area:
cm²
First, we need to find the length of the legs of the right triangle. We can use the Pythagorean theorem:
a² + b² = c²
a² + 6² = 6.5²
a² + 36 = 42.25
a² = 6.25
a = 2.5
So, one leg of the right triangle is 2.5 centimeters. Since the height of the triangle is 6 centimeters, the other leg must be:
b = √(6² - 2.5²) ≈ 5.5
Now, using this information, we can find the dimensions of the rectangle:
width = height of triangle = 6 cm
length = longer base of trapezoid - sum of lengths of legs of triangle
length = 9.5 cm - (2.5 cm + 5.5 cm) = 1.5 cm
Next, we can calculate the perimeter of the figure:
perimeter = 7 cm + 9.5 cm + 6 cm + 2.5 cm + 5.5 cm + 1.5 cm
perimeter = 32 cm
Finally, we can calculate the area of the figure:
area = area of trapezoid + area of triangle
area = [(shorter base + longer base) / 2] x height + (base x height) / 2
area = [(7 cm + 9.5 cm) / 2] x 6 cm + (2.5 cm x 6 cm) / 2
area = 48.75 cm² + 7.5 cm²
area = 56.25 cm²