1964 AHSME Problems/Problem 22
Contents
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 of , so the ratio is , or .
Solution 2
Therefore, , giving us the answer . -nullptr07
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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