The problem asks us to find the area of the trapezoid shown in the image. The trapezoid has one base of length 8 m, one leg of length 8 m forming a 45-degree angle with the longer base, and a height equal to the perpendicular distance between the two bases.

1. Problem Description

The problem asks us to find the area of the trapezoid shown in the image. The trapezoid has one base of length 8 m, one leg of length 8 m forming a 45-degree angle with the longer base, and a height equal to the perpendicular distance between the two bases.

2. Solution Steps

First, we need to find the height of the trapezoid. Since the angle between the 8 m side and the longer base is 45 degrees, and the height is perpendicular to the longer base, we can form a right triangle. In this right triangle, one angle is 45 degrees, so the other angle is also 45 degrees, making it an isosceles right triangle.

Therefore, the height of the trapezoid is equal to the length of the base of this right triangle. Since the hypotenuse of this right triangle is 8 m, we can use the properties of a 45-45-90 triangle to find the height.

Let $h$ be the height of the trapezoid, which is also the length of the base of the right triangle. Since it's an isosceles triangle, the base and height are equal. Then, by Pythagorean theorem:

$h^2 + h^2 = 8^2$

$2h^2 = 64$

$h^2 = 32$

$h = \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$ m.

Next, we need to find the length of the longer base. The longer base is equal to the sum of the shorter base (8 m) and the base of the 45-45-90 triangle, which is the height, $4\sqrt{2}$ m. So the longer base is $8 + 4\sqrt{2}$ m.

Now, we can use the formula for the area of a trapezoid:

$Area = \frac{1}{2} (base_1 + base_2) \cdot height$

Here, $base_1 = 8$ m, $base_2 = 8 + 4\sqrt{2}$ m, and $height = 4\sqrt{2}$ m.

$Area = \frac{1}{2} (8 + 8 + 4\sqrt{2}) \cdot 4\sqrt{2}$

$Area = \frac{1}{2} (16 + 4\sqrt{2}) \cdot 4\sqrt{2}$

$Area = (8 + 2\sqrt{2}) \cdot 4\sqrt{2}$

$Area = 32\sqrt{2} + 8(2)$

$Area = 32\sqrt{2} + 16$

$Area \approx 32(1.414) + 16$

$Area \approx 45.248 + 16$

$Area \approx 61.248$

Rounding to the nearest tenth, we have $Area \approx 61.2 \, m^2$.

3. Final Answer

61.2 $m^2$