An isosceles trapezoid has bases of 4 and 10. If the base angle is 45°, find the area.
21
To find the area of the isosceles trapezoid, we need to know the height of the trapezoid. The height can be calculated by using trigonometry.
Step 1: Find the length of the leg of the trapezoid using the base angle (45°) and the side adjacent to it.
- The side adjacent to the base angle is half the difference between the two bases.
- In this case, the side adjacent is (10 - 4)/2, which equals 3.
Step 2: Use trigonometry to find the height.
- The height of the trapezoid is given by the formula h = leg * tan(base angle).
- In this case, the height is 3 * tan(45°).
Step 3: Calculate the area of the trapezoid.
- The area of a trapezoid is given by the formula A = (1/2) * (sum of bases) * height.
- In this case, the area is (1/2) * (4 + 10) * (3 * tan(45°)).
Let's calculate the area using these steps:
- The length of the leg is 3.
- The height of the trapezoid is 3 * tan(45°) ≈ 3 * 1 ≈ 3.
- The area of the trapezoid is (1/2) * (4 + 10) * 3 ≈ 7 * 3 = 21.
Therefore, the area of the isosceles trapezoid is 21 square units.
To find the area of an isosceles trapezoid, you can use the formula:
Area = (1/2) * (a + b) * h
where 'a' and 'b' are the lengths of the bases, and 'h' is the height of the trapezoid.
In this case, the bases are 4 and 10, and the base angle is 45°.
To find the height of the trapezoid, you can use the sine of the base angle:
sin(45°) = h / (10 - 4)
Using the sine of 45°, which is (√2 / 2), we can substitute it into the equation:
(√2 / 2) = h / 6
Now, we can solve for the height 'h':
h = (√2 / 2) * 6
h = 6√2 / 2
h = 3√2
Now, substitute the values of 'a', 'b', and 'h' into the formula to find the area:
Area = (1/2) * (4 + 10) * 3√2
Area = (1/2) * 14 * 3√2
Area = 21√2
Therefore, the area of the isosceles trapezoid is 21√2.