2005 CEMC Gauss (Grade 7) Problems/Problem 13

Problem

In the diagram, the length of $DC$ is twice the length of $BD$ . What is the area of the triangle $ABC$ ?

$\text{(A)}\ 24 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 36$

$[asy] draw((0,0)--(-3,0)--(0,4)--cycle); draw((0,0)--(6,0)--(0,4)--cycle); label("3",(-1.5,0),N); label("4",(0,2),E); label("$A$",(0,4),N); label("$B$",(-3,0),S); label("$C$",(6,0),S); label("$D$",(0,0),S); draw((0,0.4)--(0.4,0.4)--(0.4,0)); [/asy]$

Solution

Since $BD = 3$ and $DC$ is twice the length of $BD$ , then $DC = 6$ . Therefore, triangle $ABC$ has a base of length $9$ and a height of length $4$ . Therefore, the area of triangle $ABC$ is $\frac{1}{2}bh = \frac{1}{2}(9)(4) = \frac{1}{2}(36) = 18$ . Therefore, the correct answer is $D$

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2005 CEMC Gauss (Grade 7) Problems/Problem 13

Problem

Solution

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