Shape is a trapezoid:

LINE AC = 12dm
LINE FD = 13dm
LINE BE = 14dm
LINE CG = 10dm This line makes the right angle of the triangle
What is the area?

To find the area of a trapezoid, you can use the formula:

Area = (b1 + b2) * h / 2

Where b1 and b2 are the lengths of the parallel bases and h is the height of the trapezoid.

In this case, to find the area, we need to determine the lengths of the parallel bases and the height.

From the given information, it's mentioned that LINE AC is equal to 12 dm, LINE FD is equal to 13 dm, LINE BE is equal to 14 dm, and LINE CG forms a right angle.

From this information, we can determine that the lengths of the parallel bases of the trapezoid are AC and FD, which are 12 dm and 13 dm respectively.

The height of the trapezoid can be determined by finding the length of the line CG since it forms a right angle.

Using the Pythagorean theorem, we can calculate the length of CG:

CG^2 = AC^2 - GC^2

CG^2 = 12^2 - 10^2

CG^2 = 144 - 100

CG^2 = 44

Taking the square root of both sides:

CG ≈ √44 ≈ 6.63 dm

Now that we have the lengths of the parallel bases (12 dm and 13 dm) and the height (6.63 dm), we can use the formula to find the area of the trapezoid.

Area = (b1 + b2) * h / 2

Area = (12 + 13) * 6.63 / 2

Area = 25 * 6.63 / 2

Area ≈ 82.875 dm^2

Therefore, the area of the trapezoid is approximately 82.875 square decimeters.