Midline of trapezoid and diagonals

Question

The parallel sides of a trapezoid have lengths m and n. The diagonals of the trapezoid divide the midline into three pieces. In terms of m and n, find the lengths of these pieces.

Answer 1

Draw and label a diagram!

m is the bottom base

A is the intersection of mid-line and left side

D is the intersection of mid-line and right side

B is intersection of diagonal and mid-line to the right of A

C is intersection of diagonal and mid-line to the left of D.

 

BD = m/2, AC = m/2       similar triangle

AD = (m + n) /  2           property of mid-line

AD = AC + BD - 2BC        segment addition

(m+n)/2 = m/2 + m/2 - 2BC    substitution

(m+n)/2 = m - 2BC

m/2 + n/2 = m - 2BC

-m/2 + n/2 = -2BC

m/2 - n/2 = 2BC

m/4 - n/4 = BC

 

Now you can determine AB and CD.