2021 Fall AMC 10A Problems/Problem 15
Contents
Problem
Isosceles triangle has
, and a circle with radius
is tangent to line
at
and to line
at
. What is the area of the circle that passes through vertices
,
, and
Solution 1 (Cyclic Quadrilateral)
~MRENTHUSIASM ~kante314
Solution 2 (Similar Triangles)
Because circle
is tangent to
at
. Because O is the circumcenter of
is the perpendicular bisector of
, and
, so therefore
by AA similarity. Then we have
. We also know that
because of the perpendicular bisector, so the hypotenuse of
.
This is the radius of the circumcircle of
, so the area of this circle is
Solution in Progress
~KingRavi
Solution 3 (Trigonometry)
Denote by the center of the circle that is tangent to line
at
and to line
at
.
Because this circle is tangent to line at
,
and
.
Because this circle is tangent to line
at
,
and
.
Because ,
,
,
.
Hence,
.
Let and
meet at point
.
Because
,
,
,
.
Hence, and
.
Denote . Hence,
.
Denote by the circumradius of
.
In
, following from the law of sines,
.
Therefore, the area of the circumcircle of is
Therefore, the answer is
.
~Steven Chen (www.professorchenedu.com)
Video Solution by The Power of Logic
~math2718281828459
See Also
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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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