Final answer:
To find the area of trapezoid DEFG, you can use the formula A = (DE + FG) * h / 2, where DE and FG are the bases and h is the height. By analyzing the properties of the trapezoid and using the given information, you can find the values of DE and FG in terms of 'a', and substitute them into the area formula to get the final answer in terms of 'a' and 'h'.
Step-by-step explanation:
In a trapezoid, the area can be found by multiplying the average of the bases by the height. Since DE = FG, DE and FG are the bases. Let's call the height of the trapezoid h.
The area formula for a trapezoid is A = (DE + FG) * h / 2. Given that DF = a and m<FDG = 45, we can find the value of DE and FG in terms of 'a' using the properties of a trapezoid.
Since m<FDG = 45, m<DGF is also 45 degrees. Therefore, the triangle DGF is an isosceles triangle. Let's consider the triangle DGE, which is also isosceles, with DE = FG = x (the length of the bases).
In a right triangle, if one angle is 45 degrees, the other two angles are also 45 degrees. Therefore, the other angles in triangle DGE are both 45 degrees.
Since the angles in a triangle sum to 180 degrees, the sum of the angles in triangle DGE is 45 + 45 + 90 = 180 degrees. Therefore, DGE is a right triangle.
By applying the Pythagorean theorem to triangle DGE, we can find the value of x in terms of 'a'. The Pythagorean theorem states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.
In triangle DGE, the hypotenuse is DE = x, and the other two sides are DG = a and GE = a. Applying the Pythagorean theorem, we get x^2 = a^2 + a^2 = 2a^2. Therefore, x = sqrt(2a^2) = sqrt(2)*a.
Now, we can substitute the values of DE and FG in terms of 'a' into the area formula. A = (DE + FG) * h / 2 = (2x) * h / 2 = x * h = sqrt(2)*a * h.
Therefore, the area of trapezoid DEFG is sqrt(2)*a * h.