Difference between revisions of "2013 AMC 10A Problems/Problem 18"
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<math> \textbf{(A)}\ 54\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 70\qquad\textbf{(E)}\ 75 </math> | <math> \textbf{(A)}\ 54\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 70\qquad\textbf{(E)}\ 75 </math> | ||
+ | [[Category: Introductory Geometry Problems]] | ||
==Solution== | ==Solution== |
Revision as of 11:50, 13 August 2014
Problem
Let points ,
,
, and
. Quadrilateral
is cut into equal area pieces by a line passing through
. This line intersects
at point
, where these fractions are in lowest terms. What is
?
Solution
First, various area formulas (shoelace, splitting, etc) allow us to find that . Therefore, each equal piece that the line separates
into must have an area of
.
Call the point where the line through intersects
. We know that
. Furthermore, we know that
, as
. Thus, solving for
, we find that
, so
. This gives that the y coordinate of E is
.
Line CD can be expressed as , so the
coordinate of E satisfies
. Solving for
, we find that
.
From this, we know that .
See Also
2013 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 10 Problems and Solutions |
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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.