2013 AMC 10A Problems/Problem 16
Problem
A triangle with vertices ,
, and
is reflected about the line
to create a second triangle. What is the area of the union of the two triangles?
Solution 1
Let be at
, B be at
, and
be at
. Reflecting over the line
, we see that
,
(as the x-coordinate of B is 8), and
. Line
can be represented as
, so we see that
is on line
.
We see that if we connect to
, we get a line of length
(between
and
). The area of
is equal to
.
Now, let the point of intersection between and
be
. If we can just find the area of
and subtract it from
, we are done.
We realize that because the diagram is symmetric over , the intersection of lines
and
should intersect at an x-coordinate of
. We know that the slope of
is
. Thus, we can represent the line going through
and
as
. Plugging in
, we find that the y-coordinate of F is
. Thus, the height of
is
. Using the formula for the area of a triangle, the area of
is
.
To get our final answer, we must subtract this from .
Solution 2
First, realize that is the midpoint of
and
is the midpoint of
. Connect
to
to form
. Let the midpoint of
be
. Connect
to
.
is a median of
.
Because is isosceles,
is also an altitude of
. We know the length of
and
from the given coordinates. The area of
is
.
Let the intesection of ,
and
be
.
is the centroid of
. Therefore, it splits
into
and
. The area of quadrilateral
~Zeric Hang
Solution 3
Since we have to find the area on a coordinate plane, we can use the Shoelace Theorem to find the area of the intersection. When you reflect it, it makes a quadrilateral.
Since it is reflected around , the point
remains the same on both. The top right corners are
, and its reflection
. Now to find the 4th point, point F, we can use the equation of the line DE(
, and substitute
, to get
. Now we can use the theorem:
See Also
2013 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 |
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