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Difference between revisions of "2009 AMC 12A Problems/Problem 10"

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By the [[triangle inequality]] we have <math>BD < DA + AB = 9 + 5 = 14</math>, and also <math>BD + CD > BC</math>, hence <math>BD > BC - CD = 17 - 5 = 12</math>.
 
By the [[triangle inequality]] we have <math>BD < DA + AB = 9 + 5 = 14</math>, and also <math>BD + CD > BC</math>, hence <math>BD > BC - CD = 17 - 5 = 12</math>.
  
−
We got that <math>12 < BD < 14</math>, and as we know that <math>BD</math> is an integer, we must have <math>BD=\boxed{13}</math>.
+
We get that <math>12 < BD < 14</math>, and as we know that <math>BD</math> is an integer, we must have <math>BD=\boxed{13}</math>.
  
 
== See Also ==
 
== See Also ==

Latest revision as of 08:31, 8 June 2021

The following problem is from both the 2009 AMC 12A #10 and 2009 AMC 10A #12, so both problems redirect to this page.

Problem

In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$?

[asy] unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), B=(17,0); pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0]; pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0]; pair[] dotted={A,B,C,D}; draw(D--A--B--C--D--B); dot(dotted); label("$D$",D,NW); label("$C$",C,W); label("$B$",B,E); label("$A$",A,NE); [/asy]

$\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

Solution

By the triangle inequality we have $BD < DA + AB = 9 + 5 = 14$, and also $BD + CD > BC$, hence $BD > BC - CD = 17 - 5 = 12$.

We get that $12 < BD < 14$, and as we know that $BD$ is an integer, we must have $BD=\boxed{13}$.

See Also

2009 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions
2009 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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