What is the area of the trapezoid shown below?

Hello there!

Your answer is 180 units²

Step-by-step explanation:

[tex] \large \boxed{ \mathsf{ \frac{1}{2} \times sum \: of \: two \: parallel \: \times height}}[/tex]

The formula in the box is an area of trapezoid formula. We know the sum of two parallel lines which is 15 because 7+4 = 11 + 4 = 15

another 4 comes from parallel side. If both sides are parallel, they have same length.

The only thing that is missing is the height. The height can be found by using Pythagorean Theorem as you notice a right-angle triangle when splitting in half.

[tex] \large \boxed{ {a}^{2} + {b}^{2} = {c}^{2} }[/tex]

The formula in the box is Pythagorean Theorem for finding a length of right-angle triangle.

Given where a is opposite, b is adjacent and c is hypotenuse. (Note that c must be hypotenuse.)

Our a is missing

Our b is 7

Our c is 25.

From the formula and given lengths:

[tex] \large{ {a}^{2} + {7}^{2} = {25}^{2} } \\ \large{ {a}^{2} + 49 = 625} \\ \large{ {a}^{2} = 625 - 49} \\ \large{ {a}^{2} = 576} \\ \large{a = 24}[/tex]

Therefore, our opposite is 24. Since our opposite equals the height of a trapezoid. We can proceed with the trapezoid formula.

[tex] \large{ \frac{1}{2} \times 15 \times 24 = 15 \times 12 = 180 } \\ \large{180}[/tex]

Therefore, the area of trapezoid is 180 units²

We can also use another method to find the area by adding up between area of triangle and area of Rectangle.

Our rectangle formula is length × width. We know width which is 4 and our length is 24 from opposite side of triangle which equal to the length of rectangle.

Length × Width = 24 × 4 = 96

Hence, the area of Rectangle is 96.

Next, we find the area of triangle which is 1/2 × base × height.

Our base is 7 and height is the opposite side of triangle which is 24.

Therefore, the area of triangle is 1/2 × 7 × 24 = 7 × 12 = 84

If we add the area of triangle and rectangle up each other, we will get the area of trapezoid for this problem which is 96 from rectangle and 84 from triangle. Therefore, 96+84 = 180 units²

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