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.
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
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