Difference between revisions of "1996 AJHSME Problems/Problem 17"
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* [[AJHSME Problems and Solutions]] | * [[AJHSME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
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Revision as of 23:25, 4 July 2013
Problem
Figure 
is a square. Point 
is the origin, and point 
has coordinates (2,2). What are the coordinates for 
so that the area of triangle 
equals the area of square ?
![[asy] pair O,P,Q,R,T; O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0); draw((-5,0)--(3,0)); draw((0,-1)--(0,3)); draw(P--Q--R); draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8)); draw((-0.2,2.8)--(0,3)--(0.2,2.8)); draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2)); draw((2.8,-0.2)--(3,0)--(2.8,0.2)); draw(Q--T); label("$O$",O,SW); label("$P$",P,S); label("$Q$",Q,NE); label("$R$",R,W); label("$T$",T,S); [/asy]](http://latex.artofproblemsolving.com/e/9/f/e9f1cf12b73280352cc045ff945ace2d4228d2f9.png)
Solution
The area of 
is 
.
The area of 
is 
.
If we set the areas equal, the area of $\traingle PQT$ (Error compiling LaTeX. Unknown error_msg) is 
. Also, note that 
. Plugging those in, we get:
If , and 
, then 
, and must be 
units to the left of the origin. This would be 
, giving answer 
.
See Also
| 1996 AJHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
