Difference between revisions of "2020 AMC 12A Problems/Problem 17"
Siluweston (talk | contribs) (→Solution 2) |
Siluweston (talk | contribs) (→solution 3) |
||
Line 28: | Line 28: | ||
~Solution by IronicNinja | ~Solution by IronicNinja | ||
− | |||
− | |||
==See Also== | ==See Also== |
Revision as of 12:06, 2 February 2020
Contents
Problem 17
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 left-most -coordinate be
Recall that, by the shoelace formula, the area of the triangle must be That equals to
The -coordinate is, therefore,
~lopkiloinm.
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
See Also
2020 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.