2014 AMC 10A Problems/Problem 16
Contents
Problem
In rectangle , , , and points , , and are midpoints of , , and , respectively. Point is the midpoint of . What is the area of the shaded region?
Solution
Solution 1
Note that the region is a kite; hence its diagonals are perpendicular and it has area for diagonals of length and . Since as both and are midpoints of parallel sides of rectangle and , we let . Now all we need to do is to find .
Let the other two vertices of the kite be and with closer to than . This gives us . Now let . We thus find that the equation of is and that of is . Solving this system gives us , so the -coordinate of is ; in other words, is from . By symmetry, is also the same distance from , so as we have . Hence the area of the kite is .
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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