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Difference between revisions of "2006 AMC 8 Problems/Problem 19"

(Created page with "== Problem == Triangle <math> ABC </math> is an isosceles triangle with <math> \overline{AB}=\overline{BC}</math>. Point <math> D </math> is the midpoint of both <math> \overlin...")
(No difference)

Revision as of 22:45, 14 November 2011

Problem

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 $\overline{BD}$?

[asy] size(100); draw((0,0)--(2,4)--(4,0)--(6,4)--cycle--(4,0),linewidth(1)); label("$A$", (0,0), SW); label("$B$", (2,4), N); label("$C$", (4,0), SE); label("$D$", shift(0.2,0.1)*intersectionpoint((0,0)--(6,4),(2,4)--(4,0)), N); label("$E$", (6,4), NE);[/asy]

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 4.5\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 5.5\qquad\textbf{(E)}\ 6$

Solution

Since triangle $ABD$ is congruent to triangle $ECD$ and $\overline{CE} =11$, $\overline{AB}=11$. Since $\overline{AB}=\overline{BC}$, $\overline{BC}=11$. Because point $D$ is the midpoint of $\overline{BC}$, $\overline{BD}=\frac{\overline{BC}}{2}=\frac{11}{2}=\boxed{\textbf{(D)}\ 5.5}$.

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