Difference between revisions of "2009 AMC 12B Problems/Problem 9"
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=== Solution 4 === | === Solution 4 === | ||
| − | WLOG, let the coordinates of <math>C</math> be <math>(3,4)</math> , or any coordinate, for that matter. Applying the shoelace formula, we get the area as <math>\boxed 6</math> | + | WLOG, let the coordinates of <math>C</math> be <math>(3,4)</math> , or any coordinate, for that matter. Applying the shoelace formula, we get the area as <math>\boxed 6</math>. |
| + | |||
| + | ~coolmath2017 | ||
== See also == | == See also == | ||
Latest revision as of 20:19, 27 April 2020
Contents
Problem
Triangle 
has vertices 
, 
, and 
, where is on the line 
. What is the area of 
?
Solution
Solution 1
Because the line is parallel to 
, the area of is independent of the location of on that line. Therefore it may be assumed that is 
. In that case the triangle has base 
and altitude 
, so its area is 
.
Solution 2
The base of the triangle is 
. Its altitude is the distance between the point 
and the parallel line , which is 
. Therefore its area is 
. The answer is 
.
![[asy] unitsize(0.75cm); defaultpen(0.8); pair A=(3,0), B=(0,3); draw ( (-1,0) -- (9,0), dashed ); draw ( (0,-1) -- (0,9), dashed ); dot(A); dot(B); draw(A--B); draw ( (-1,8) -- (8,-1) ); label( "$A$", A, S ); label( "$B$", B, W ); label( "$3$", A--(0,0), S ); label( "$3$", B--(0,0), W ); label( "$x+y=7$", (8,-1), SE ); pair C = intersectionpoint(A--(10,7),(7,0)--(0,7)); draw( A--C, dashed ); draw(rightanglemark(A,C,(7,0))); draw(rightanglemark(C,A,B)); label( "$4$", A--(7,0), S ); label( "$3\sqrt 2$", 0.67*B+0.33*A, NE ); label( "$\frac 4{\sqrt 2}$", A--C, NW ); label( "$\frac 4{\sqrt 2}$", C--(7,0), NE ); [/asy]](http://latex.artofproblemsolving.com/f/a/b/fab711cac413774cb38c162f63d0bf76374d2421.png)
Solution 3
By Shoelace, our area is:
![\[\frac {1}{2} \cdot |(3\cdot 3 + 0 \cdot y+ 0 \cdot x)-(0*0+3(x+y))|.\]](http://latex.artofproblemsolving.com/1/3/9/1396c23f943303e97f684936ca70a7a56859b487.png)
We know 
so we get:
![\[\frac {1}{2} \cdot |9-21|=\boxed 6\]](http://latex.artofproblemsolving.com/2/0/7/20734e7f08db10f3e04d7cef82ab45630ef51ce4.png)
Solution 4
WLOG, let the coordinates of be 
, or any coordinate, for that matter. Applying the shoelace formula, we get the area as 
.
~coolmath2017
See also
| 2009 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 8 |
Followed by Problem 10 |
| 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 AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
