1964 AHSME Problems/Problem 22
Problem
Given parallelogram with the midpoint of diagonal . Point is connected to a point in so that . What is the ratio of the area of to the area of quadrilateral ?
Solution
If it works for a parallelogram , it should also work for a unit square, with . We are given that is the midpoint of , so . If is on , then . We note that and , so means , or , and hence .
We note that has a base that is and an altitude from to that is . Therefore, .
Quadrilateral can be split into and . The first triangle is of the unit square cut diagonally, so . The second triangle has base that is and height to that is . Therefore, .
The entire quadrilateral has area . This is times larger than the area if , so the ratio is , or .
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.