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Problem 238: (2009 AMC 10a, 12; 2009 AMC12a 10)
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Similar Triangles
areas
pythagoreanTheorem
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Problem 238: (2009 AMC 10a, 12; 2009 AMC12a 10)
In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$?
$\mathrm{(A)}\ 11 \qquad$ $ \mathrm{(B)}\ 12 \qquad$ $ \mathrm{(C)}\ 13 \qquad$ $ \mathrm{(D)}\ 14 \qquad$ $ \mathrm{(E)}\ 15$
Solution
Triangle inequality in $\triangle ABD$ ($AB=5,DA=9$): $4<BD<14$.
Triangle inequality in $\triangle BCD$ ($BC=17,CD=5$): $12<BD<22$.
The overlap is $12<BD<14$, so $BD=13$.
The answer is
(C)
.
Exercise 238:
Solution
In quadrilateral $ABCD$, $AB = 6$, $BC = 13$, $CD = 6$, $DA = 3$, and $BD$ is an integer. What is $BD$?
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$\mathrm{(A)}\ 11 \qquad$ $ \mathrm{(B)}\ 12 \qquad$ $ \mathrm{(C)}\ 13 \qquad$ $ \mathrm{(D)}\ 14 \qquad$ $ \mathrm{(E)}\ 15$ 
