2015 AMC 12B Problems/Problem 19
Contents
Problem
In ,
and
. Squares
and
are constructed outside of the triangle. The points
,
,
, and
lie on a circle. What is the perimeter of the triangle?
Solution 1
First, we should find the center and radius of this circle. We can find the center by drawing the perpendicular bisectors of and
and finding their intersection point. This point happens to be the midpoint of
, the hypotenuse. Let this point be
. To find the radius, determine
, where
,
, and
. Thus, the radius
.
Next we let and
. Consider the right triangle
first. Using the pythagorean theorem, we find that
. Next, we let
to be the midpoint of
, and we consider right triangle
. By the pythagorean theorem, we have that
. Expanding this equation, we get that
This means that is a 45-45-90 triangle, so
. Thus the perimeter is
which is answer
.
image needed
Solution 2
The center of the circle on which ,
,
, and
lie must be equidistant from each of these four points. Draw the perpendicular bisectors of
and of
. Note that the perpendicular bisector of
is parallel to
and passes through the midpoint of
. Therefore, the triangle that is formed by
, the midpoint of
, and the point at which this perpendicular bisector intersects
must be similar to
, and the ratio of a side of the smaller triangle to a side of
is 1:2. Consequently, the perpendicular bisector of
passes through the midpoint of
. The perpendicular bisector of
must include the midpoint of
as well. Since all points on a perpendicular bisector of any two points
and
are equidistant from
and
, the center of the circle must be the midpoint of
.
Now the distance between the midpoint of and
, which is equal to the radius of this circle, is
. Let
. Then the distance between the midpoint of
and
, also equal to the radius of the circle, is given by
(the ratio of the similar triangles is involved here). Squaring these two expressions for the radius and equating the results, we have
Since cannot be equal to 12, the length of the hypotenuse of the right triangle, we can divide by
, and arrive at
. The length of other leg of the triangle must be
. Thus, the perimeter of the triangle is
.
See Also
2015 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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 |
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