2020 AMC 12A Problems/Problem 17
Contents
Problem
The vertices of a quadrilateral lie on the graph of , and the -coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is . What is the -coordinate of the leftmost vertex?
Solution 1
Let the coordinates of the quadrilateral be . We have by shoelace's theorem, that the area is We now that the numerator must have a factor of , so given the answer choices, is either or . If , the expression does not evaluate to , but if , the expression evaluates to . Hence, our answer is .
Solution 2
Like above, use the shoelace formula to find that the area of the triangle is equal to . Because the final area we are looking for is , the numerator factors into and , which one of and has to be a multiple of and the other has to be a multiple of . Clearly, the only choice for that is
~Solution by IronicNinja
Solution 3
How is a concave function, then:
Therefore , all quadrilaterals of side right are trapezius
~Solution by AsdrúbalBeltrán
Video Solution by TheBeautyofMath
https://www.youtube.com/watch?v=Eq2A2TTahqU?t=583 Another example of shoelace theorem included earlier in the video
~IceMatrix
See Also
2020 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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All AMC 12 Problems and Solutions |
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