Difference between revisions of "2005 AIME I Problems/Problem 10"
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== Problem == | == Problem == | ||
| − | [[Triangle]] <math> ABC </math> lies in the [[Cartesian Plane]] and has an [[area]] of 70. The coordinates of <math> B </math> and <math> C </math> are <math> (12,19) </math> and <math> (23,20), </math> respectively, and the coordinates of <math> A </math> are <math> (p,q). </math> The [[line]] containing the [[median]] to side <math> BC </math> has [[slope]] <math> -5. </math> Find the largest possible value of <math> p+q. </math> | + | [[Triangle]] <math> ABC </math> lies in the [[Cartesian Plane]] and has an [[area]] of 70. The coordinates of <math> B </math> and <math> C </math> are <math> (12,19) </math> and <math> (23,20), </math> respectively, and the coordinates of <math> A </math> are <math> (p,q). </math> The [[line]] containing the [[median of a triangle | median]] to side <math> BC </math> has [[slope]] <math> -5. </math> Find the largest possible value of <math> p+q. </math> |
== Solution == | == Solution == | ||
Revision as of 13:22, 14 February 2007
Problem
Triangle 
lies in the Cartesian Plane and has an area of 70. The coordinates of 
and 
are 
and 
respectively, and the coordinates of 
are 
The line containing the median to side 
has slope 
Find the largest possible value of 
Solution
The midpoint 
of line segment 
is 
. Let 
be the point 
, which lies along the line through of slope 
. The area of triangle 
can be computed in a number of ways (one possibility: extend 
until it hits the line 
, and subtract one triangle from another), and each such calculation gives an area of 14. This is 
of our needed area, so we simply need the point to be 5 times as far from as is. Thus 
, and the sum of coordinates will be larger if we take the positive value, so 
and the answer is 
.
