Proving Midpoints of a diagonal are on the Mid segment of a trapazoid

Question

Given trapezoid ABCD with bases AB and CD, draw diagonals AC and BD. Let E be the midpoint of AC and F the midpoint of BD.

 

Prove that E and F lie on the mid segment of the trapazoid

 

If AB = 10 and DC = 22 Find the lenght of EF

Answer 1

EF = AB + DC

       -----------

             2

 

EF = 10 + 22

      ------------

           2

 

EF = 16

 

 

Draw the trapezoid marking the information given.  Draw the height of the trapezoid. Note that you have two 45 degree angles in the triangle portion of the trapezoid. The base of the triangle is 12. (22-10, 10 is the base of the rectangle or remaining portion of the trapezoid).  Since one side of the 45 Degree angle is 12, the other side opposite of the 45 degree angle must be 12 (the height).  EF divides the triangle into a large and small triangle. So the height of the larger triangle can be divided resulting in 6 (above segment EF) and 6 (below EF).  Looking at the smaller triangle we see that the side opposite the 45 degree angle is also 6.  Now we have the 2 parts of EF, 10 and 6 which equals 16.