Difference between revisions of "2000 PMWC Problems/Problem T3"
(→Problem) |
Megaboy6679 (talk | contribs) m |
||
| (One intermediate revision by the same user not shown) | |||
| 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>. | ||
| + | |||
| + | ~megaboy6679 | ||
==See Also== | ==See Also== | ||
Latest revision as of 18:59, 30 January 2023
Problem
In the figure, 
is a square, 
, 
, 
, and 
are midpoints of the sides 
, 
, 
and 
respectively. Find the ratio of the shaded area to the area of the square .
![[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]](http://latex.artofproblemsolving.com/b/a/a/baa5783b2f8d6973e8fe88bcc8ecbfb56857698e.png)
Solution
Rotate the four smallest triangles around their corresponding midpoints, so that there will be 
squares, which 
is shaded. Since the area didn't change, the ratio of the shaded area to the area of square ABCD is 
.
~megaboy6679
