A trapezoid has an area of 144 square meters and an altitude of 4 meters. Its two bases have a ratio of 4:5. What is the perimeter of the trapezoid?

Question

A trapezoid has an area of 144 square meters and an altitude of 4 meters.  Its two bases have a ratio of 4:5. What is the perimeter of the trapezoid?

Caroline Aucoin · Accepted Answer

#x + g(x) + 72#

#b/B = 4/5#

#A = (B + b)/2 cdot h#

#144 = (B + 4/5 B)/2 cdot 4#

#72 = B + 4/5 B = 9/5 B Rightarrow B = 5/9 cdot 72 = 40#

#b = 4/5 cdot 40 = 32#

#B - b = 8#

There are infinitely many #x#'s , #y#'s such that the area of triangle is #1/2 cdot 8 cdot 4 = 16#

Heron's formula: #2p = x + y + 8#

#16 = sqrt{p(p-x)(p-y)(p-8)} = f(x, y)#

We want trapezoid's perimeter = #x + y + 40 + 32#

Theodore Abad · Answer

undefined To find the perimeter of the trapezoid, we first need to determine the lengths of its bases. We can use the formula for the area of a trapezoid, which is given by:

Area = (1/2) * (sum of the lengths of the bases) * altitude

Given that the area is 144 square meters and the altitude is 4 meters, we can plug these values into the formula and solve for the sum of the lengths of the bases.

144 = (1/2) * (4 + x) * 4

Where 'x' represents the length of the longer base.

Solving for 'x':

144 = 2 * (4 + x)
72 = 4 + x
x = 68

So, the longer base is 68 meters.

Since the ratio of the bases is 4:5, we can find the length of the shorter base by setting up the proportion:

4/x = 4/5

Cross-multiplying:

4 * 5 = 4 * x
20 = 4x
x = 5

So, the shorter base is 5 meters.

Now, we can find the perimeter of the trapezoid by adding the lengths of all four sides:

Perimeter = shorter base + longer base + 2 * height
Perimeter = 5 + 68 + 2 * 4
Perimeter = 5 + 68 + 8
Perimeter = 81 meters.