Difference between revisions of "2006 AMC 10B Problems/Problem 23"
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Revision as of 22:57, 10 May 2011
Problem
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7 as shown. What is the area of the shaded quadrilateral?
Solution
Label the points in the figure as shown below, and draw the segment . This segment divides the quadrilateral into two triangles, let their areas be and .
Since triangles and share an altitude from and have equal area, their bases must be equal, hence .
Since triangles and share an altitude from and their respective bases are equal, their areas must be equal, hence .
Since triangles and share an altitude from and their respective areas are in the ratio , their bases must be in the same ratio, hence .
Since triangles and share an altitude from and their respective bases are in the ratio , their areas must be in the same ratio, hence , which gives us .
Substituting into the second equation we get , which solves to . Then , and the total area of the quadrilateral is .
See also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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 |