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 21:11, 16 April 2009
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Problem
Isosceles has a right angle at
. Point
is inside
, such that
,
, and
. Legs
and
have length $s=\sqrt{a+b\sqrt{2}{$ (Error compiling LaTeX. Unknown error_msg), where
and
are positive integers. What is
?
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 |