Final answer:
To find the area of the isosceles trapezoid, we can use the Pythagorean theorem to find the height. Then, we can use the formula for the area of a trapezoid to calculate the area. The area of the isosceles trapezoid is approximately 141.4 square units.
Step-by-step explanation:
To find the area of the isosceles trapezoid, we need to know the lengths of the parallel sides and the height. The longest leg is given as 10 units, and the hypotenuse is given as 15 units. We also know that the measure of the angle closest to the hypotenuse is 40º. Since it is an isosceles trapezoid, the two legs opposite the equal angles are congruent.
To find the height, we can use the Pythagorean theorem. Let's consider the right triangle formed by one of the legs, the height, and the hypotenuse. Using the Pythagorean theorem, we have (leg)² + (height)² = (hypotenuse)². Substituting the given values, we get (10/2)² + (height)² = 15². Simplifying this equation gives us 25 + (height)² = 225. Solving for height, we find (height)² = 200. Taking the square root of both sides, we get height ≈ 14.14 units.
Now that we have the lengths of the parallel sides (10 units and 10 units) and the height (14.14 units), we can calculate the area of the trapezoid. The formula for the area of a trapezoid is (base1 + base2) * height / 2. Substituting the given values, we get (10 + 10) * 14.14 / 2 = 20 * 14.14 / 2 = 141.4 square units.