Find the area of the trapezoid. Leave your answer in simplest radical form. The figure is not drawn to scale. A trapezoid is shown. Its shorter base is at the top of the image and its length is 12 centimeters. The trapezoid’s height is 6 centimeters. Vertical dotted lines are drawn through the trapezoid dividing it into two right triangles and a rectangle. The triangle on the left side of the trapezoid has an angle at the bottom left corner labeled 45 degrees. The triangle on the right side of the trapezoid has its horizontal side at the bottom of the triangle labeled 2 centimeters. (1 point) Responses 84 cm2 84 cm 2 96 cm2 96 cm 2 72 cm2 72 cm 2 108 cm2 108 cm 2

Answer 1

To find the area of the trapezoid, we need to first find the length of the longer base. We can do this by using trigonometry in the right triangle on the right side of the trapezoid.

Since the angle at the bottom right corner of the right triangle is 45 degrees and the horizontal side at the bottom of the triangle is 2 centimeters, we can use the sine function to find the length of the longer base:

sin(45 degrees) = opposite/hypotenuse
sin(45 degrees) = 6/b (where b is the longer base)

Solving for b, we get:
b = 6/sin(45 degrees)
b = 6/sqrt(2)
b = 3sqrt(2)

Now that we have the lengths of both bases and the height, we can find the area of the trapezoid using the formula:

Area = (1/2) x (sum of bases) x height
Area = (1/2) x (12 + 3sqrt(2)) x 6
Area = (1/2) x (12 + 3sqrt(2)) x 6
Area = (1/2) x (72 + 18sqrt(2))
Area = 36 + 9sqrt(2)

Therefore, the area of the trapezoid is 36 + 9sqrt(2) square centimeters.