Question

the bottom of an isosceles trapezoid is 3 times the length of the top. if the midsegment (aka median) is 28 inches, what are the lengths of the top and bottom of the trapezoid?
top =

bottom =

Explanation:

Step1: Recall the midsegment formula for a trapezoid

The midsegment (median) \( m \) of a trapezoid is the average of the lengths of the two bases (top and bottom). The formula is \( m=\frac{a + b}{2} \), where \( a \) is the length of the top base and \( b \) is the length of the bottom base.

Step2: Define variables

Let the length of the top base be \( x \). Then the length of the bottom base is \( 3x \) (since the bottom is 3 times the top). The midsegment \( m = 28 \) inches.

Step3: Substitute into the formula

Substitute \( a=x \), \( b = 3x \), and \( m = 28 \) into the formula \( m=\frac{a + b}{2} \). We get \( 28=\frac{x+3x}{2} \).

Step4: Simplify and solve for \( x \)

First, simplify the numerator: \( x + 3x=4x \). So the equation becomes \( 28=\frac{4x}{2} \). Simplify \( \frac{4x}{2}=2x \). Now we have \( 28 = 2x \). Divide both sides by 2: \( x=\frac{28}{2}=14 \).

Step5: Find the bottom base length

Since the bottom base is \( 3x \), substitute \( x = 14 \): \( 3x=3\times14 = 42 \).

Answer:

top = 14 inches
bottom = 42 inches