Difference between revisions of "1997 PMWC Problems/Problem T1"
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Triangles UWQ, PUY, UWX, and UXY are all right triangles with side lengths 1, <math>\sqrt{3}</math>, and 2. Thus <math>[UWXY]=\sqrt{3}</math> and <math>[PQR]=\frac{9}{4}\sqrt{3}</math>. <math>\frac{[UWXY]}{[PQR]}=\frac{1}{\frac{9}{4}}=\boxed{\frac{4}{9}}</math> | Triangles UWQ, PUY, UWX, and UXY are all right triangles with side lengths 1, <math>\sqrt{3}</math>, and 2. Thus <math>[UWXY]=\sqrt{3}</math> and <math>[PQR]=\frac{9}{4}\sqrt{3}</math>. <math>\frac{[UWXY]}{[PQR]}=\frac{1}{\frac{9}{4}}=\boxed{\frac{4}{9}}</math> | ||
| − | ==See | + | ==See Also== |
{{PMWC box|year=1997|num-b=I15|num-a=T2}} | {{PMWC box|year=1997|num-b=I15|num-a=T2}} | ||
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| + | [[Category:Introductory Geometry Problems]] | ||
Revision as of 15:06, 15 May 2012
Problem
Let 
be an equilateral triangle with sides of length three units. 
, 
, 
, 
, 
, and 
divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral 
to the area of the triangle .
Solution
Triangles UWQ, PUY, UWX, and UXY are all right triangles with side lengths 1, 
, and 2. Thus ![$[UWXY]=\sqrt{3}$](http://latex.artofproblemsolving.com/7/4/b/74b218e3575edf1a5a282ea57a1c4c964b5b81f9.png)
and ![$[PQR]=\frac{9}{4}\sqrt{3}$](http://latex.artofproblemsolving.com/e/8/0/e8071a13673274b9441560234a3f66cc1b1252a6.png)
. ![$\frac{[UWXY]}{[PQR]}=\frac{1}{\frac{9}{4}}=\boxed{\frac{4}{9}}$](http://latex.artofproblemsolving.com/5/2/4/5245bd7efa2fd8bb67b886baf83dc146324a9bf8.png)
See Also
| 1997 PMWC (Problems) | ||
| Preceded by Problem I15 |
Followed by Problem T2 | |
| 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 | ||
