Trapezoid question

The question:

"ABCD is a trapezoid with the measure of base AB twice the measure of the base CD. Point E is the point of intersection of the diagonals. The measure of diagonal AC is 11. Find the length of segment EC. Express your answer as a common fraction."

∠AEB  and  ∠DEC  are vertical angles, so they are congruent.

∠CAB  and  ∠ACD  are alternate angles, so they are congruent.

By the AA similarity theorem, △ABE ~ △CDE, so...

AE / AB   =   CE / CD

The measure of diagonal AC is 11.

AE + CE  =  AC

AE + CE  =  11

AE  =  11 - CE

And the measure of base AB is twice the measure of the base CD.

AB  =  2CD

Now we can use these values of  AE  and  AB .

AE / AB  =  CE / CD

                                                     Substitute  2CD  in for  AB  and  11 - CE  in for  AE.

(11 - CE) / (2CD)  =  CE / CD

                                                     Multiply both sides of the equation by  CD .

(11 - CE) / 2  =  CE

                                                     Multiply both sides of the equation by  2 .

11 - CE   =   2CE

                                                     Add  CE  to both sides.

11  =  3CE

                                                     Divide both sides by  3 .

11 / 3  =  CE

EC  =  11 / 3

Hint: ABE and CDE are similar when drawing the diagram.