2009 AMC 10A Problem 12

Attempt Problem 12 of the 2009 AMC 10A 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 10A solutions, or check the answer key.

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12.

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

separately

The diagonal

BD

must satisfy

12 \lt BD \lt 14

\triangle BCD,

the triangle inequality gives

5 + BD \gt 17,

BD \gt 12.

\triangle ABD,

it gives

5 + 9 \gt BD,

BD \lt 14.

The only integer with

12 \lt BD \lt 14

BD = 13.

Thus, the correct answer is C.

Problem 12 in other years