Question

I need to use the formula for a trapezoid and find the area and perimeter

Answer 1

Hello there. To solve this question, we'll have to remember some properties about trapezoids and right triangles.

Given the following trapezoid:

We have to determine its area and perimeter.

For this, remember that:

The area of a trapezoid with bases B and b (larger and smaller, respectively) and height h can be found by the formula

$A=\dfrac{(B+b)\cdot h}{2}$

The perimeter is the sum of the measures of all sides of the figure.

For the perimeter, we'll use the pythagorean theorem to determine the measure of the legs of the trapezoid.

Okay. Notice that in the trapezoid, the larger base B measures 41, the smaller base measures 21 and the height is 18.

By the formula for area, we get

$A=\dfrac{(41+21)\cdot18}{2}=62\cdot9=558$

Now, notice we can determine a right triangle on the left:

To determine the legs of the triangle, we make

$\dfrac{41-21}{2}=\dfrac{20}{2}=10$

Now we have a right triangle with legs 10 and 18.

Using the Pythagorean theorem:

$a^2+b^2=c^2$

For a triangle with legs a and b and hypotenuse c, the sum of the squares of the legs is equal to the square of the hypotenuse.

Using a = 10 and b = 18, we get

$\begin{gathered} 10^2+18^2=c^2 \\ 100+324=c^2 \\ 424=c^2 \\ 4\cdot106=c^2 \\ c=2\sqrt{106} \end{gathered}$

Since the other right triangle is the same, the other leg have the same measure, hence we add

$\text{ Perimeter }=21+41+2\cdot2\sqrt{106}=62+4\sqrt{106}$

We can approximate this value using a calculator

$\text{ Perimeter }\approx103.18$