AoPSWiki

Trying to get to the USAMO in 2010? Our AIME Problem Series can help you get there! Click here to enroll today!

Personal tools

Search

Toolbox

2006 AMC 12B Problems/Problem 23

From AoPSWiki

Revision as of 02:00, 17 April 2009 by M1sterzer0 (Talk | contribs)

(diff) ← Older revision | Current revision (diff) | Newer revision → (diff)

This is an empty template page which needs to be filled. You can help us out by finding the needed content and editing it in. Thanks.

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 $s=\sqrt{a+b\sqrt{2}{$ , where $a$ and $b$ are positive integers. What is $a+b$ ?

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

Solution

Using the Law of Cosines on $\triangle PBC$ , we have:

$\begin{align*}PB^2&=BC^2+PC^2-2\cdot BC\cdot PC\cdot \cos(\alpha) \Rightarrow 49 = 36 + s^2 - 12s\cos(\alpha) \Rightarrow...$

Using the Law of Cosines on $\triangle PAC$ , we have: $\begin{align*}PA^2&=AC^2+PC^2-2\cdot AC\cdot PC\cdot \cos(90^\circ-\alpha) \Rightarrow 121 = 36 + s^2 - 12s\sin(\alpha) \...$

Now we use $\sin^2(\alpha) + \cos^2(\alpha) = 1$ . $\begin{align*}\sin^2(\alpha)+\cos^2(\alpha) = 1 &\Rightarrow \frac{s^4-20s^2+169}{144s^2} + \frac{s^4-170s^2+7225}{144s^2...$

Note that we know that we want the solution with $s^2 > 85$ since we know that $\sin(\alpha) > 0$ . Thus, $a+b=85+42=\boxed{127}$ .

See also

2006 AMC 12B (Problems • 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

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/2006_AMC_12B_Problems/Problem_23"

Category: Empty template page

Want to learn how to tackle those tough AMC/AIME/Olympiad algebra problems? Check out Art of Problem Solving's Intermediate Algebra by Richard Rusczyk and Mathew Crawford. Over 1600 problems!

© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us