Difference between revisions of "1997 PMWC Problems/Problem T1"
(New page: ==Problem== Let <math>PQR</math> be an equilateral triangle with sides of length three units. <math>U</math>, <math>V</math>, <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</m...) |
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Let <math>PQR</math> be an equilateral triangle with sides of length three units. <math>U</math>, <math>V</math>, <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math> divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral <math>UWXY</math> to the area of the triangle <math>PQR</math>. | Let <math>PQR</math> be an equilateral triangle with sides of length three units. <math>U</math>, <math>V</math>, <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math> divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral <math>UWXY</math> to the area of the triangle <math>PQR</math>. | ||
| − | + | <asy> | |
| + | draw((1/2,0)--(-1/2,0)--(0,sqrt(3)/2)--cycle); | ||
| + | dot((1/6,sqrt(3)/3)); | ||
| + | dot((-1/6,sqrt(3)/3)); | ||
| + | dot((1/3,sqrt(3)/6)); | ||
| + | dot((-1/3,sqrt(3)/6)); | ||
| + | dot((1/6,0)); | ||
| + | dot((1/6,0)); | ||
| + | dot((-1/6,0)); | ||
| + | filldraw((-1/6,sqrt(3)/3)--(1/3,sqrt(3)/6)--(1/6,0)--(-1/6,0)--cycle); | ||
| + | label("$P$",(0,sqrt(3)/2),N); | ||
| + | label("$Z$",(1/6,sqrt(3)/3),NE); | ||
| + | label("$Y$",(1/3,sqrt(3)/6),NE); | ||
| + | label("$R$",(1/2,0),E); | ||
| + | label("$X$",(1/6,0),S); | ||
| + | label("$W$",(-1/6,0),S); | ||
| + | label("$Q$",(-1/2,0),W); | ||
| + | label("$V$",(-1/3,sqrt(3)/6),NW); | ||
| + | label("$U$",(-1/6,sqrt(3)/3),NW); | ||
| + | //Credit to chezbgone2 for the diagram</asy> | ||
==Solution== | ==Solution== | ||
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}} | ||
| + | |||
| + | [[Category:Introductory Geometry Problems]] | ||
Latest revision as of 13:38, 20 April 2014
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 .
![[asy] draw((1/2,0)--(-1/2,0)--(0,sqrt(3)/2)--cycle); dot((1/6,sqrt(3)/3)); dot((-1/6,sqrt(3)/3)); dot((1/3,sqrt(3)/6)); dot((-1/3,sqrt(3)/6)); dot((1/6,0)); dot((1/6,0)); dot((-1/6,0)); filldraw((-1/6,sqrt(3)/3)--(1/3,sqrt(3)/6)--(1/6,0)--(-1/6,0)--cycle); label("$P$",(0,sqrt(3)/2),N); label("$Z$",(1/6,sqrt(3)/3),NE); label("$Y$",(1/3,sqrt(3)/6),NE); label("$R$",(1/2,0),E); label("$X$",(1/6,0),S); label("$W$",(-1/6,0),S); label("$Q$",(-1/2,0),W); label("$V$",(-1/3,sqrt(3)/6),NW); label("$U$",(-1/6,sqrt(3)/3),NW); //Credit to chezbgone2 for the diagram[/asy]](http://latex.artofproblemsolving.com/d/f/8/df8e04731c7bede69fa531f84b04ac6f4baae3cc.png)
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 | ||
