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Difference between revisions of "2022 SSMO Team Round Problems/Problem 3"

(Created page with "==Problem== Let <math>ABCD</math> be an isosceles trapezoid such that <math>AB\parallel CD.</math> Let <math>E</math> be a point on <math>CD</math> such that <math>AB=CE.</mat...")
 
 
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==Problem==
 
==Problem==
 
Let <math>ABCD</math> be an isosceles trapezoid such that <math>AB\parallel CD.</math> Let <math>E</math> be a point on <math>CD</math> such that <math>AB=CE.</math> Let the midpoint of <math>DE</math> be <math>M</math> such that <math>BD</math> intersects <math>AM</math> at <math>G</math> and <math>AE</math> at <math>F.</math> If <math>DC=36, AB=24,</math> and <math>AD=10,</math> then <math>[AGF]</math> can be expressed as <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math>
 
Let <math>ABCD</math> be an isosceles trapezoid such that <math>AB\parallel CD.</math> Let <math>E</math> be a point on <math>CD</math> such that <math>AB=CE.</math> Let the midpoint of <math>DE</math> be <math>M</math> such that <math>BD</math> intersects <math>AM</math> at <math>G</math> and <math>AE</math> at <math>F.</math> If <math>DC=36, AB=24,</math> and <math>AD=10,</math> then <math>[AGF]</math> can be expressed as <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math>
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+
<center>
−
[asy] unitsize(2mm); fill((6,8/5)--(10,8/3)--(6,8)--cycle,lightgray); draw((0,0)--(36,0)--(30,8)--(6,8)--cycle); draw((6,8)--(12,0)--(0,0)); draw((0,0)--(30,8)); draw((6,8)--(6,0)); label("A", (6,8), NW); dot((6,8)); label("B", (30,8), NE); dot((30,8)); label("C", (36,0), SE); dot((36,0)); label("D", (0,0), SW); dot((0,0)); label("E",(12,0),S); dot((12,0)); label("M",(6,0),S); dot((6,0)); dot((6,8/5)); dot((10,8/3)); [/asy]
+
<asy>
 +
unitsize(2mm);
 +
fill((6,8/5)--(10,8/3)--(6,8)--cycle,lightgray);
 +
draw((0,0)--(36,0)--(30,8)--(6,8)--cycle);
 +
draw((6,8)--(12,0)--(0,0));
 +
draw((0,0)--(30,8));
 +
draw((6,8)--(6,0));
 +
label("A", (6,8), NW);
 +
dot((6,8));
 +
label("B", (30,8), NE);
 +
dot((30,8));
 +
label("C", (36,0), SE);
 +
dot((36,0));
 +
label("D", (0,0), SW);
 +
dot((0,0));
 +
label("E",(12,0),S);
 +
dot((12,0));
 +
label("M",(6,0),S);
 +
dot((6,0));
 +
dot((6,8/5));
 +
dot((10,8/3));
 +
</asy>
 +
</center>
  
 
==Solution==
 
==Solution==

Latest revision as of 12:08, 14 December 2023

Problem

Let $ABCD$ be an isosceles trapezoid such that $AB\parallel CD.$ Let $E$ be a point on $CD$ such that $AB=CE.$ Let the midpoint of $DE$ be $M$ such that $BD$ intersects $AM$ at $G$ and $AE$ at $F.$ If $DC=36, AB=24,$ and $AD=10,$ then $[AGF]$ can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

[asy] unitsize(2mm); fill((6,8/5)--(10,8/3)--(6,8)--cycle,lightgray); draw((0,0)--(36,0)--(30,8)--(6,8)--cycle); draw((6,8)--(12,0)--(0,0)); draw((0,0)--(30,8)); draw((6,8)--(6,0)); label("A", (6,8), NW); dot((6,8)); label("B", (30,8), NE); dot((30,8)); label("C", (36,0), SE); dot((36,0)); label("D", (0,0), SW); dot((0,0)); label("E",(12,0),S); dot((12,0)); label("M",(6,0),S); dot((6,0)); dot((6,8/5)); dot((10,8/3)); [/asy]

Solution

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