To find the area of an isosceles trapezoid with bases of lengths 4 and 10, and a base angle of 45 degrees, we can break the problem down into steps.
First, we need to understand the geometry of the trapezoid. An isosceles trapezoid has two parallel sides (the bases) and the non-parallel sides (the legs) are equal in length. The formula for the area of a trapezoid is:
Area = 0.5 × (Base1 + Base2) × Height
In our case, Base1 (the shorter base) is 4 and Base2 (the longer base) is 10.
Next, we need to determine the height of the trapezoid. We can visualize dropping perpendiculars from the endpoints of the shorter base to the longer base, creating right triangles at both ends. The angle at the base of these triangles is 45 degrees, and we can use trigonometric relationships to find the height.
Since the base angle is 45 degrees, we know that for a 45-45-90 triangle, the lengths of the legs are equal. The difference between the lengths of the bases is:
Difference = Base2 – Base1 = 10 – 4 = 6
Since there are two right triangles, the horizontal distance from the endpoint of the shorter base to where the perpendicular meets the longer base is half of this difference:
Horizontal Distance = Difference/2 = 6/2 = 3
Now we can find the height. In a 45-degree triangle:
Height = Horizontal Distance = 3
Now that we have the height, we can calculate the area:
Area = 0.5 × (4 + 10) × 3
Area = 0.5 × 14 × 3
Area = 21
Thus, the area of the isosceles trapezoid is 21 square units.