Difference between revisions of "2009 AMC 10B Problems/Problem 16"
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== Problem == | == Problem == | ||
− | Points <math>A</math> and <math>C</math> lie on a circle centered at <math>O</math> | + | Points <math>A</math> and <math>C</math> lie on a circle centered at <math>O</math>, each of <math>\overline{BA}</math> and <math>\overline{BC}</math> are tangent to the circle, and <math>\triangle ABC</math> is equilateral. The circle intersects <math>\overline{BO}</math> at <math>D</math>. What is <math>\frac{BD}{BO}</math>? |
<math> | <math> |
Revision as of 19:35, 12 January 2021
Problem
Points and
lie on a circle centered at
, each of
and
are tangent to the circle, and
is equilateral. The circle intersects
at
. What is
?
Solution
Solution 1
As is equilateral, we have
, hence
. Then
, and from symmetry we have
. Thus, this gives us
.
We know that , as
lies on the circle. From
we also have
, Hence
, therefore
, and
.
Solution 2
As in the previous solution, we find out that . Hence
and
are both equilateral.
We then have , hence
is the incenter of
, and as
is equilateral,
is also its centroid. Hence
, and as
, we have
, therefore
, and as before we conclude that
.
See Also
2009 AMC 10B (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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