2016 AMC 12B Problems/Problem 10
Problem
A quadrilateral has vertices , , , and , where and are integers with . The area of is . What is ?
Solution
Note that the slope of is and the slope of is . Hence, and we can similarly prove that the other angles are right angles. This means that is a rectangle. By distance formula we have . Simplifying we get . Thus and have to be a factor of 8. The only way for them to be factors of and remain integers is if and . So the answer is
Solution by I_Dont_Do_Math
Solution 2
Solution by e_power_pi_times_i
By the Shoelace Theorem, the area of the quadrilateral is , so . Since and are integers, and , so .
Video Solution by TheBeautyofMath
https://www.youtube.com/watch?v=Eq2A2TTahqU with a second example of Shoelace Theorem done after this problem
~IceMatrix
See Also
2016 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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All AMC 12 Problems and Solutions |
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