Difference between revisions of "2009 AMC 10B Problems/Problem 18"
(New page: == Problem == Rectangle <math>ABCD</math> has <math>AB=8</math> and <math>BC=6</math>. Point <math>M</math> is the midpoint of diagonal <math>\overline{AC}</math>, and <math>E</math> is o...) |
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Revision as of 19:46, 8 March 2009
Problem
Rectangle has and . Point is the midpoint of diagonal , and is on with . What is the area of ?
Solution
By the Pythagorean theorem we have , hence .
The triangles and have the same angle at and a right angle, thus all their angles are equal, and therefore these two triangles are similar.
The ratio of their sides is , hence the ratio of their areas is .
And as the area of triangle is , the area of triangle is .
See Also
2009 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 10 Problems and Solutions |