Difference between revisions of "2003 AIME II Problems/Problem 6"
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== Problem == | == Problem == | ||
− | In triangle <math>ABC,</math> <math>AB = 13,</math> <math>BC = 14,</math> <math>AC = 15,</math> and point <math>G</math> is the intersection of the medians. Points <math>A',</math> <math>B',</math> and <math>C',</math> are the images of <math>A,</math> <math>B,</math> and <math>C,</math> respectively, after a <math>180^\circ</math> rotation about <math>G.</math> What is the area | + | In triangle <math>ABC,</math> <math>AB = 13,</math> <math>BC = 14,</math> <math>AC = 15,</math> and point <math>G</math> is the intersection of the medians. Points <math>A',</math> <math>B',</math> and <math>C',</math> are the images of <math>A,</math> <math>B,</math> and <math>C,</math> respectively, after a <math>180^\circ</math> rotation about <math>G.</math> What is the area of the union of the two regions enclosed by the triangles <math>ABC</math> and <math>A'B'C'?</math> |
== Solution == | == Solution == |
Revision as of 23:07, 14 March 2009
Problem
In triangle and point is the intersection of the medians. Points and are the images of and respectively, after a rotation about What is the area of the union of the two regions enclosed by the triangles and
Solution
Since a triangle is a triangle and a triangle "glued" together on the side, .
There are six points of intersection between and . Connect each of these points to .
There are smaller congruent triangles which make up the desired area. Also, is made up of of such triangles. Therefore, .
See also
2003 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |