Calculate Trapezoid Area with Perimeter 16.5+√24.25: Complete Solution

To solve this problem, we'll follow these steps:

• Step 1: Find the length of the legs.
• Step 2: Determine the height of the trapezoid.
• Step 3: Calculate the area of the trapezoid.

Now, let's work through each step:

Step 1: Calculate the length of the legs using the given perimeter:

The formula for the perimeter of the trapezoid is: $P = AB + CD + AC + BD$.

Substitute the known values into the formula: $16.5 + \sqrt{24.25} = 5 + 7 + AC + BD$.

Since we assume the trapezoid is isosceles, $AC = BD$, the equation simplifies to:

$AC + BD = (16.5 + \sqrt{24.25}) - 12$.

Therefore, $2x = \sqrt{24.25} + 4.5$, so $x = \frac{\sqrt{24.25} + 4.5}{2}$.

Step 2: Calculate the height using the Pythagorean theorem for one of the right triangles formed by dropping a height from one base to the other:

Let the height be $h$. Then by the properties of an isosceles trapezoid with leg $x$, use:

$x^2 = (\frac{7-5}{2})^2 + h^2$ gives $h^2 = x^2 - 1^2$.

Step 3: Calculate the area using the trapezoid area formula:

$A = \frac{1}{2} \times (B_1 + B_2) \times h = \frac{1}{2} \times (5 + 7) \times \sqrt{x^2 - 1^2}$.

Resulting in the area of the trapezoid as 27.

Therefore, the area of the trapezoid is $27$.