Difference between revisions of "1997 PMWC Problems/Problem I5"
(New page: ==Problem== Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas? Image:1997 PMWC individual 5.png ==Solution...) |
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Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas? | Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas? | ||
| − | + | <asy> | |
| + | import patterns; | ||
| + | /* modified Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ | ||
| + | import graph; usepackage("amsmath"); size(3.95cm); | ||
| + | real labelscalefactor = 0.5; /* changes label-to-point distance */ | ||
| + | pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ | ||
| + | pen dotstyle = black; /* point style */ | ||
| + | real xmin = -6.01, xmax = 15.94, ymin = -3.31, ymax = 13.18; /* image dimensions */ | ||
| + | draw((3.94,4.18)--(6,5.09)--(6,6)--(3.14,6)--cycle); | ||
| + | /* draw figures */ | ||
| + | draw((0,0)--(0,6)); | ||
| + | draw((0,0)--(6,0)); | ||
| + | draw((0,6)--(6,6)); | ||
| + | draw((6,6)--(6,0)); | ||
| + | draw((2.33,7.85)--(3.94,4.18)); | ||
| + | draw((2.33,7.85)--(5.99,9.45)); | ||
| + | draw((5.99,9.45)--(7.6,5.79)); | ||
| + | draw((7.6,5.79)--(3.94,4.18)); | ||
| + | label("$ 6\text{cm} $",(-1.45,2.86),fontsize(15)); | ||
| + | label("$ 4\text{cm} $",(8.46,7.95),fontsize(15)); | ||
| + | add("crosshatch",crosshatch(.7mm)); | ||
| + | fill((6,5.09)--(3.94,4.18)--(3.14,6)--(6,6)--cycle, pattern("crosshatch")); | ||
| + | /* dots and labels */ | ||
| + | clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); | ||
| + | //Credit to dasobson for the diagram</asy> | ||
==Solution== | ==Solution== | ||
| Line 13: | Line 37: | ||
<math>(A+C)-(B+C)=A-B=\boxed{20}</math> | <math>(A+C)-(B+C)=A-B=\boxed{20}</math> | ||
| − | ==See | + | ==See Also== |
| + | {{PMWC box|year=1997|num-b=I3|num-a=I6}} | ||
Latest revision as of 13:20, 20 April 2014
Problem
Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas?
![[asy] import patterns; /* modified Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; usepackage("amsmath"); size(3.95cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -6.01, xmax = 15.94, ymin = -3.31, ymax = 13.18; /* image dimensions */ draw((3.94,4.18)--(6,5.09)--(6,6)--(3.14,6)--cycle); /* draw figures */ draw((0,0)--(0,6)); draw((0,0)--(6,0)); draw((0,6)--(6,6)); draw((6,6)--(6,0)); draw((2.33,7.85)--(3.94,4.18)); draw((2.33,7.85)--(5.99,9.45)); draw((5.99,9.45)--(7.6,5.79)); draw((7.6,5.79)--(3.94,4.18)); label("$ 6\text{cm} $",(-1.45,2.86),fontsize(15)); label("$ 4\text{cm} $",(8.46,7.95),fontsize(15)); add("crosshatch",crosshatch(.7mm)); fill((6,5.09)--(3.94,4.18)--(3.14,6)--(6,6)--cycle, pattern("crosshatch")); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); //Credit to dasobson for the diagram[/asy]](http://latex.artofproblemsolving.com/f/4/a/f4ae1ade0e1368e4ba4826e931762efd459554db.png)
Solution
Let the area of the non-shaded region of the 6 by 6 square by A. Let the area of the non-shaded region of the 4 by 4 square by B. Let the shaded area be C.
See Also
| 1997 PMWC (Problems) | ||
| Preceded by Problem I3 |
Followed by Problem I6 | |
| I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
