The equal sides of an isosceles trapezoid each measure 5, and its altitude measures 4. If the area of the trapezoid is 48, how do you find the lengths of its bases?
To find the lengths of the bases of the isosceles trapezoid, we can use the formula for the area of a trapezoid:
Area = (1/2) * (a + b) * h
where "a" and "b" are the lengths of the bases, and "h" is the altitude.
Given that the area is 48 and the altitude is 4, we can substitute these values into the formula:
48 = (1/2) * (a + b) * 4
Now, solve this equation for the sum of the bases (a + b):
48 = 2 * (a + b)
Divide both sides of the equation by 2:
24 = a + b
Since the isosceles trapezoid has equal sides measuring 5, we know that a = b = 5. Substitute these values into the equation:
24 = 5 + b
Subtract 5 from both sides of the equation:
19 = b
Therefore, the length of one base is 5 and the length of the other base is 19.
To find the lengths of the bases of the isosceles trapezoid, we can use the formula for the area of a trapezoid:
Area = ((b1 + b2) * h) / 2,
where b1 and b2 are the lengths of the bases, and h is the altitude of the trapezoid.
Given that the isosceles trapezoid has equal sides measuring 5 and an altitude of 4, and the area is 48, we can plug in these values into the formula and solve for the bases:
48 = ((b1 + b2) * 4) / 2.
To eliminate the fraction, we can multiply both sides of the equation by 2:
96 = (b1 + b2) * 4.
Now, divide both sides by 4:
24 = b1 + b2.
Since the trapezoid has equal sides, we can denote one base as x, and the other base as x + 10 (5 + 5), as the length of the equal sides is 5. So, we have:
24 = x + (x + 10).
Simplifying the equation:
24 = 2x + 10.
Subtracting 10 from both sides:
14 = 2x.
Dividing both sides by 2:
7 = x.
Therefore, one base of the isosceles trapezoid is 7, and the other base is x + 10 = 7 + 10 = 17.
area = average of the bases * altitude
average of the bases = 48 / 4 = 12
... so the sum of the bases is 24
the equal sides and the altitude
... form right triangles at the ends of the longer base
... two 3-4-5 triangles means that the longer base is 6 longer
b + b + 6 = 24 ... 2 b = 18 ... b = 9
so the bases are 9 and 15