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Question
Find the area of the trapezoid by decomposing it into other shapes.
Responses
A 48 cm2
48 cm 2
B 55 cm2
55 cm 2
C 66 cm2
66 cm 2
D 96 cm2
96 cm 2
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To find the area of the trapezoid, we can decompose it into a rectangle and two right triangles.
Let's label the trapezoid as ABCD, with side AB parallel to side CD. Let's also label the points where the diagonals intersect as E and F.
We can see that triangle ABE and triangle CDF are congruent right triangles, so we only need to find the area of one of them and double it.
The area of a right triangle is given by the formula 1/2 * base * height.
The base of each triangle is the length of the trapezoid, which is 13 cm.
To find the height of the triangle, we can use the Pythagorean theorem. The legs of the right triangle are the height of the trapezoid, which is 5 cm, and the difference in the lengths of the parallel sides, which is 4 cm.
Using the Pythagorean theorem, we can calculate the height of the triangle as follows:
height^2 = (5 cm)^2 - (4 cm)^2
height^2 = 25 cm^2 - 16 cm^2
height^2 = 9 cm^2
height = 3 cm
Now we can calculate the area of one triangle:
area of triangle ABE = 1/2 * 13 cm * 3 cm = 19.5 cm^2
Since there are two congruent triangles, the total area of the triangles is 2 * 19.5 cm^2 = 39 cm^2.
To find the area of the rectangle, we can use the formula length * width. The length is the height of the trapezoid, which is 5 cm, and the width is the length of the trapezoid, which is 13 cm.
area of rectangle = 5 cm * 13 cm = 65 cm^2
Finally, we can find the total area of the trapezoid by adding the areas of the triangles and the rectangle:
total area of trapezoid = area of triangles + area of rectangle
total area of trapezoid = 39 cm^2 + 65 cm^2 = 104 cm^2
Therefore, the correct answer is not listed among the given options.