Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 8"

m
(→‎Problem)
 
Line 1: Line 1:
 
==Problem==
 
==Problem==
−
Let <math>ABCDE</math> be a convex pentagon with <math>AB\sqrt{2}=BC=CD=DE</math>, <math>\angle ABC=150^\circ</math>, <math>\angle BCD=75^\circ</math>, and <math>\angle CDE=165^\circ</math>. If <math>\angle ABE=\frac{m}{n}^\circ</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>.
+
Let <math>ABCDE</math> be a convex pentagon with <math>\frac{AB}{\sqrt{2}}=BC=CD=DE</math>, <math>\angle ABC=150^\circ</math>, <math>\angle BCD=75^\circ</math>, and <math>\angle CDE=165^\circ</math>. If <math>\angle ABE=\frac{m}{n}^\circ</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>.
  
 
==Solution==
 
==Solution==

Latest revision as of 14:54, 21 August 2020

Problem

Let $ABCDE$ be a convex pentagon with $\frac{AB}{\sqrt{2}}=BC=CD=DE$, $\angle ABC=150^\circ$, $\angle BCD=75^\circ$, and $\angle CDE=165^\circ$. If $\angle ABE=\frac{m}{n}^\circ$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Solution

[asy]defaultpen(fontsize(8)); pair A=expi(pi*5/12)+expi(0)+expi(pi/2), B=expi(pi*5/12), C=(0,0), D=expi(0), E=expi(0)+expi(pi/12), P=expi(pi*5/12)+expi(0); draw(A--B--C--D--E--A);draw(B--P--E--B);draw(D--P--A); label("A",A,(1,0));label("B",B,(-1,0));label("C",C,(-1,0));label("D",D,(1,-1)); label("E",E,(1,0));label("P",P,(1,0)); dot(A^^B^^C^^D^^E^^P);[/asy]

Let $P$ be a point in $ABCDE$ such that $\angle ABP=\angle BAP=45^\circ$. We see that $\angle CBP=115^\circ$ and thus $BP||CD$. Since $BP=BC=CD$, we have that $BCDP$ is a rhombus. Therefore $\angle CDP=115^\circ$ so $\angle PDE=60^\circ$. Since $PD=CD=DE$ we have that $\triangle PDE$ is equilateral.

$\angle BPE=75+60^\circ=180^\circ-2\angle PBE\implies \angle PBE=\frac{45}{2}$. $\angle ABE=\angle ABP+\angle PBE=45+\frac{45}{2}^\circ=\frac{135}{2}^\circ\implies m+n=\boxed{137}$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Mock_AIME_1_2006-2007_Problems/Problem_8&oldid=132298"