An isosceles trapezoid has base angles equal to 45 and bases of lengths 6 and 12. Find the area of the trapezoid.
To find the area of an isosceles trapezoid, we can use the formula:
Area = (1/2) * (sum of the lengths of bases) * (height)
In this case, the bases have lengths 6 and 12, and the height is still unknown. Let's denote the height of the trapezoid as 'h'.
Since the trapezoid is isosceles, the altitude from the top vertex of the trapezoid to the base creates two right triangles with each base as the hypotenuse. The base angles of the trapezoid are equal to 45 degrees, so each right triangle in the isosceles trapezoid is a 45-45-90 triangle.
In a 45-45-90 triangle, the legs are congruent and the hypotenuse is equal to the length of one leg multiplied by the square root of 2.
Using this information, we can find the height of the trapezoid:
h = Length of one leg * square root of 2
h = 6 * sqrt(2)
h = 6 * 1.414 (using a calculator)
h ≈ 8.485
Now that we have the height, we can calculate the area:
Area = (1/2) * (sum of the lengths of bases) * (height)
Area = (1/2) * (6 + 12) * 8.485
Area = (1/2) * 18 * 8.485
Area ≈ 76.18
Therefore, the area of the isosceles trapezoid is approximately 76.18 square units.
To find the area of an isosceles trapezoid, you can use the formula:
Area = (1/2) * (sum of the lengths of the bases) * (height)
In this case, the lengths of the bases are 6 and 12, and the height is unknown. However, we can find the height by using trigonometry and the given base angles of 45 degrees.
Since the base angles are equal, we can drop a perpendicular from one of the base angles to the opposite base, forming two right triangles. Let's call the height of the trapezoid "h."
In the right triangle, with one leg equal to "h" and the other leg equal to 6/2 = 3 (half of one of the bases), we can use trigonometry to find the value of "h."
Using the tangent function:
tan(45°) = h / 3
tan(45°) = 1
h / 3 = 1
h = 3
Now that we know the height of the trapezoid is 3, we can plug all the values into the formula to find the area.
Area = (1/2) * (6 + 12) * 3
Area = (1/2) * 18 * 3
Area = 9 * 3
Area = 27
Therefore, the area of the isosceles trapezoid is 27 square units.
h = (B1 - B2) / 2 = (12 - 6)/2 = 3.
A = (B1+B2)h / 2 = (12+6)3 / 2 = 27.