2009 AMC 12A Problem 10

Attempt Problem 10 of the 2009 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12A solutions, or check the answer key.

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10.

In quadrilateral

ABCD,

AB = 5,

BC = 17,

CD = 5,

DA = 9,

and

BD

is an integer. What is

BD?

11

12

13

14

15

Apply the triangle inequality to

\triangle BCD

and

\triangle ABD

From

\triangle BCD,

BD \gt 12,

and from

\triangle ABD,

BD \lt 14

\triangle BCD,

the triangle inequality gives

BD + CD \gt BC,

BD + 5 \gt 17

and

BD \gt 12.

\triangle ABD,

AB + DA \gt BD,

BD \lt 5 + 9 = 14.

The only integer with

12 \lt BD \lt 14

13.

Thus, the correct answer is C.

Problem 10 in other years