Points A, B, C, and D are all on segment AD, in the order listed. Segment AB is congruent to segment CD. Segment BC equals 12 units in length. Also, an additional point E exists that is not on segment AD with segment BE being congruent to segment CE. Segment BE equals 10 units in length. If the perimeter of triangle ADE is twice the perimeter of perimeter of triangle BCE, find the length of segment AB.

Source: mathcontest.olemiss.edu 1/22/2007

**SOLUTION**

Since E is equidistant from B and C, E is on the perpendicular bisector . Because , M is also the midpoint of . Since is the perpendicular bisector of , E is equidistant from A and D and .

**Length of ME**

In right triangle ,

substitute for and for

subtract from both sides

**Perimeter of triangle BCE**

**Perimeter of triangle ADE**

substitute for

substitute for and for

Because the perimeter of triangle ADE is twice the perimeter of triangle BCE,

divide both sides by 2

(1)

**Length of AB**

In right triangle ,

(2) substitute for and for

Substitute the value of from Eq. (1) into Eq. (2),

subtract from both sides

apply identity

simplify

divide both sides by

**Answer**: 9

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## About mvtrinh

Retired high school math teacher.