Calculate Trapezoid Area: 22cm Perimeter Isosceles Problem

To solve this problem, we'll approach it step by step:

• Step 1: Calculate the length $X$ of the non-parallel sides.
• Step 2: Determine the height of the trapezoid.
• Step 3: Apply the area formula for trapezoids.

Let's work through these steps:

Step 1: Calculate the length $X$

The perimeter of the trapezoid is given as $22$ cm. The perimeter equation for our trapezoid $ABCD$ is:

$P = AB + CD + 2X = 22$

Substituting the given lengths, we have:

$4 + 8 + 2X = 22$

$12 + 2X = 22$

Solving for $X$, we get:

$2X = 10$

$X = 5$

Step 2: Determine the height $h$

Because the trapezoid is isosceles, we can drop perpendicular heights from the endpoints of the shorter base $AB$ to the longer base $CD$, creating right triangles at each end.

The distance between these projections on $CD$ will be $CD - AB = 8 - 4 = 4$ cm. Each of these segments will then be half this, so $2$ cm each (since the trapezoid is symmetric).

Using the Pythagorean theorem in one of these right triangles, where $X$ is the hypotenuse, and one leg is $2$, gives us:

$h^2 + 2^2 = 5^2$

$h^2 + 4 = 25$

$h^2 = 21$

$h = \sqrt{21}$

Step 3: Calculate the area using trapezoid area formula

Use the formula for the area of a trapezoid:

$\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h = \frac{1}{2} \times (4 + 8) \times \sqrt{21}$

$\text{Area} = \frac{1}{2} \times 12 \times \sqrt{21}$

$\text{Area} = 6 \times \sqrt{21}$

Therefore, the area of the trapezoid is $6 \times \sqrt{21}$.