Problem 19 · 2006 AMC 8
Medium
Geometry & Measurement
congruent-trianglesmidpoint
Triangle ABC is an isosceles triangle with AB = BC. Point D is the midpoint of both BC and AE, and CE is 11 units long. Triangle ABD is congruent to triangle ECD. What is the length of BD?
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Answer: D — 5.5.
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Hint 1 of 2
You want BD, but the only length given is CE = 11. Build a chain of equal lengths that connects them — that's what the congruence and the isosceles condition are for.
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Hint 2 of 2
Congruent triangles have equal matching sides: read ▵ABD ≅ ▵ECD in order, so AB matches EC. Match the letters position by position to find which sides are equal.
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Approach: chain equal segments from the known length to BD
- Match the congruence letter-for-letter: ▵ABD ≅ ▵ECD pairs AB with EC, so AB = EC = 11.
- The triangle is isosceles with AB = BC, so BC = 11 too.
- D is the midpoint of BC, so BD = ½ × 11 = 5.5.
- The skill to keep: never read a congruence as a blob — line the letters up in order so corresponding parts (A↔E, B↔C, D↔D) tell you exactly which sides are equal. From there it's just hopping along equal segments to the one you want.
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