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Difference between revisions of "2000 PMWC Problems/Problem T3"

(Created page with "==Problem== In the figure, <math>ABCD</math> is a square, <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> are midpoints of the sides <math>AB</math>, <math>BC<...")
 
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==Problem==
 
==Problem==
 
In the figure, <math>ABCD</math> is a square, <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> are midpoints of the sides <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively. Find the ratio of the shaded area to the area of the square <math>ABCD</math>.  
 
In the figure, <math>ABCD</math> is a square, <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> are midpoints of the sides <math>AB</math>, <math>BC</math>, <math>CD</math> and <math>DA</math> respectively. Find the ratio of the shaded area to the area of the square <math>ABCD</math>.  
 +
 +
<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);
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label("B",(2,2),NE);
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label("C",(2,0),SE);
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label("D",(0,0),SW);
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label("P",(1,2),N);
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label("Q",(2,1),E);
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label("R",(1,0),S);
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label("S",(0,1),W);
 +
//Credit to chezbgone2 for the diagram</asy>
  
 
==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>.
 +
 +
~megaboy6679
  
 
==See Also==
 
==See Also==

Latest revision as of 18:59, 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 $\boxed{\frac{1}{5}}$.

~megaboy6679

See Also

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