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Difference between revisions of "2006 AMC 12B Problems/Problem 23"

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== Problem ==
 
== Problem ==
 
Isosceles <math>\triangle ABC</math> has a right angle at <math>C</math>.  Point <math>P</math> is inside <math>\triangle ABC</math>, such that <math>PA=11</math>, <math>PB=7</math>, and <math>PC=6</math>. Legs <math>\overline{AC}</math> and <math>\overline{BC}</math> have length <math>s=\sqrt{a+b\sqrt{2}{</math>, where <math>a</math> and <math>b</math> are positive integers.  What is <math>a+b</math>?
 
Isosceles <math>\triangle ABC</math> has a right angle at <math>C</math>.  Point <math>P</math> is inside <math>\triangle ABC</math>, such that <math>PA=11</math>, <math>PB=7</math>, and <math>PC=6</math>. Legs <math>\overline{AC}</math> and <math>\overline{BC}</math> have length <math>s=\sqrt{a+b\sqrt{2}{</math>, where <math>a</math> and <math>b</math> are positive integers.  What is <math>a+b</math>?
 +
 +
<asy>
 +
pathpen = linewidth(0.7);
 +
pen f = fontsize(10);
 +
size(5cm);
 +
pair B = (0,sqrt(85+42*sqrt(2)));
 +
pair A = (B.y,0);
 +
pair C = (0,0);
 +
pair P = IP(arc(B,7,180,360),arc(C,6,0,90));
 +
D(A--B--C--cycle);
 +
D(P--A);
 +
D(P--B);
 +
D(P--C);
 +
MP("A",D(A),plain.E,f);
 +
MP("B",D(B),plain.N,f);
 +
MP("C",D(C),plain.SW,f);
 +
MP("P",D(P),plain.NE,f);
 +
</asy>
  
 
<math>
 
<math>

Revision as of 20:11, 16 April 2009

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Problem

Isosceles $\triangle ABC$ has a right angle at $C$. Point $P$ is inside $\triangle ABC$, such that $PA=11$, $PB=7$, and $PC=6$. Legs $\overline{AC}$ and $\overline{BC}$ have length $s=\sqrt{a+b\sqrt{2}{$ (Error compiling LaTeX. Unknown error_msg), where $a$ and $b$ are positive integers. What is $a+b$?

[asy] pathpen = linewidth(0.7); pen f = fontsize(10); size(5cm); pair B = (0,sqrt(85+42*sqrt(2))); pair A = (B.y,0); pair C = (0,0); pair P = IP(arc(B,7,180,360),arc(C,6,0,90)); D(A--B--C--cycle); D(P--A); D(P--B); D(P--C); MP("A",D(A),plain.E,f); MP("B",D(B),plain.N,f); MP("C",D(C),plain.SW,f); MP("P",D(P),plain.NE,f); [/asy]

$\mathrm{(A)}\ 85 \qquad \mathrm{(B)}\ 91 \qquad \mathrm{(C)}\ 108 \qquad \mathrm{(D)}\ 121 \qquad \mathrm{(E)}\ 127$

Solution

See also

2006 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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
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