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 "2000 PMWC Problems/Problem T3"

(→‎Problem)
m
Line 20: Line 20:
  
 
==Solution==
 
==Solution==
 +
Rotate the four smallest triangles around their corresponding midpoints, so that there will be <math>5</math> squares, which <math>1</math> is shaded. Since the area didn't change, the ratio of the shaded area to the area of square ABCD is <math>\boxed{\frac{1}{5}}</math>
  
 
==See Also==
 
==See Also==

Revision as of 18:57, 30 January 2023

Problem

In the figure, $ABCD$ is a square, $P$, $Q$, $R$, and $S$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ respectively. Find the ratio of the shaded area to the area of the square $ABCD$.

[asy] fill(buildcycle((0,2)--(1,0),(2,2)--(0,1),(2,0)--(1,2),(0,0)--(2,1)),gray); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((0,2)--(1,0)); draw((2,2)--(0,1)); draw((2,0)--(1,2)); draw((0,0)--(2,1)); label("A",(0,2),NW); label("B",(2,2),NE); label("C",(2,0),SE); label("D",(0,0),SW); label("P",(1,2),N); label("Q",(2,1),E); label("R",(1,0),S); label("S",(0,1),W); //Credit to chezbgone2 for the diagram[/asy]

Solution

Rotate the four smallest triangles around their corresponding midpoints, so that there will be $5$ squares, which $1$ is shaded. Since the area didn't change, the ratio of the shaded area to the area of square ABCD is $\boxed{\frac{1}{5}}$

See Also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2000_PMWC_Problems/Problem_T3&oldid=188415"