2006 AMC 8 Problem 19

Below is the professionally curated solution for Problem 19 of the 2006 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 8 solutions, or check the answer key.

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Concepts:congruence (geometry)isosceles triangle midpoint

Difficulty rating: 1410

19.

Triangle $ABC$ is an isosceles triangle with $\overline{AB}=\overline{BC}.$ Point $D$ is the midpoint of both $\overline{BC}$ and $\overline{AE},$ and $\overline{CE}$ is $11$ units long. Triangle $ABD$ is congruent to triangle $ECD.$ What is the length of $\overline{BD}?$

$4$

$4.5$

$5$

$5.5$

$6$

Solution:

By the congruency condition, we know that $AB = EC = 11.$

Also from the isosceles condition, we know that $BC = AB = 11.$

Since $D$ is the midpoint of $\overline{BC},$ we know that $BD = BC \div 2 = 5.5$

Thus, D is the correct answer.

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