Final answer:
To find the area of the isosceles trapezoid, we can use the Pythagorean theorem to find the height. Once we have the height, we can use the formula for the area of a trapezoid to calculate the area.
Step-by-step explanation:
To find the area of an isosceles trapezoid, we need to know the lengths of the two parallel sides (a and b) and the height (h). In this case, we have side lengths 8 cm and 4 cm, and the height is not given. However, we can use the Pythagorean theorem to find the height. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Since the trapezoid is isosceles, we can draw a line from one of the vertices perpendicular to the base to form a right triangle. Let's call the length of this line x.
Using the Pythagorean theorem, we have:
x^2 + h^2 = 8^2
x^2 + h^2 = 64
Since the trapezoid is isosceles, we know that x is equal to the length of the other base (4 cm).
Substituting x = 4 into the equation, we have:
4^2 + h^2 = 64
16 + h^2 = 64
h^2 = 48
h = sqrt(48)
h ≈ 6.93 cm
Now that we know the height, we can calculate the area of the trapezoid:
Area = 1/2 * (a + b) * h
Area = 1/2 * (8 + 4) * 6.93
Area ≈ 34.65 cm²