Difference between revisions of "2013 AMC 10A Problems/Problem 23"
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[[Category: Introductory Geometry Problems]] | [[Category: Introductory Geometry Problems]] | ||
− | + | ==Solution 1 (Power of a Point)== | |
Let <math>BX = q</math>, <math>CX = p</math>, and <math>AC</math> meets the circle at <math>Y</math> and <math>Z</math>, with <math>Y</math> on <math>AC</math>. Then <math>AZ = AY = 86</math>. Using the Power of a Point (Secant-Secant Power Theorem), we get that <math>p(p+q) = 11(183) = 11 * 3 * 61</math>. We know that <math>p+q>p</math>, so <math>p</math> is either 3,11, or 33. We also know that <math>p>11</math> by the triangle inequality on <math>\triangle ACX</math>. Thus, <math>p</math> is <math>33</math> so we get that <math>BC = p+q = \boxed{\textbf{(D) }61}</math>. | Let <math>BX = q</math>, <math>CX = p</math>, and <math>AC</math> meets the circle at <math>Y</math> and <math>Z</math>, with <math>Y</math> on <math>AC</math>. Then <math>AZ = AY = 86</math>. Using the Power of a Point (Secant-Secant Power Theorem), we get that <math>p(p+q) = 11(183) = 11 * 3 * 61</math>. We know that <math>p+q>p</math>, so <math>p</math> is either 3,11, or 33. We also know that <math>p>11</math> by the triangle inequality on <math>\triangle ACX</math>. Thus, <math>p</math> is <math>33</math> so we get that <math>BC = p+q = \boxed{\textbf{(D) }61}</math>. | ||
− | + | ==Solution 2 ([[Stewart's Theorem]])== | |
Let <math>x</math> represent <math>CX</math>, and let <math>y</math> represent <math>BX</math>. Since the circle goes through <math>B</math> and <math>X</math>, <math>AB = AX = 86</math>. | Let <math>x</math> represent <math>CX</math>, and let <math>y</math> represent <math>BX</math>. Since the circle goes through <math>B</math> and <math>X</math>, <math>AB = AX = 86</math>. | ||
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The prime factors of <math>2013</math> are <math>3</math>, <math>11</math>, and <math>61</math>. Obviously, <math>x < x+y</math>. In addition, by the Triangle Inequality, <math>BC < AB + AC</math>, so <math>x+y < 183</math>. Therefore, <math>x</math> must equal <math>33</math>, and <math>x+y</math> must equal <math> \boxed{\textbf{(D) }61}</math>. | The prime factors of <math>2013</math> are <math>3</math>, <math>11</math>, and <math>61</math>. Obviously, <math>x < x+y</math>. In addition, by the Triangle Inequality, <math>BC < AB + AC</math>, so <math>x+y < 183</math>. Therefore, <math>x</math> must equal <math>33</math>, and <math>x+y</math> must equal <math> \boxed{\textbf{(D) }61}</math>. | ||
− | + | ==Solution 3== | |
Let <math>CX=x, BX=y</math>. Let the circle intersect <math>AC</math> at <math>D</math> and the diameter including <math>AD</math> intersect the circle again at <math>E</math>. | Let <math>CX=x, BX=y</math>. Let the circle intersect <math>AC</math> at <math>D</math> and the diameter including <math>AD</math> intersect the circle again at <math>E</math>. | ||
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Therefore, the answer is <math> \boxed{\textbf{(D) }61}</math>. | Therefore, the answer is <math> \boxed{\textbf{(D) }61}</math>. | ||
− | + | ==Solution 4== | |
<asy> | <asy> | ||
unitsize(2); | unitsize(2); |
Revision as of 20:58, 28 December 2019
- The following problem is from both the 2013 AMC 12A #19 and 2013 AMC 10A #23, so both problems redirect to this page.
Contents
[hide]Problem
In ,
, and
. A circle with center
and radius
intersects
at points
and
. Moreover
and
have integer lengths. What is
?
Solution 1 (Power of a Point)
Let ,
, and
meets the circle at
and
, with
on
. Then
. Using the Power of a Point (Secant-Secant Power Theorem), we get that
. We know that
, so
is either 3,11, or 33. We also know that
by the triangle inequality on
. Thus,
is
so we get that
.
Solution 2 (Stewart's Theorem)
Let represent
, and let
represent
. Since the circle goes through
and
,
.
Then by Stewart's Theorem,
(Since cannot be equal to
, dividing both sides of the equation by
is allowed.)
The prime factors of are
,
, and
. Obviously,
. In addition, by the Triangle Inequality,
, so
. Therefore,
must equal
, and
must equal
.
Solution 3
Let . Let the circle intersect
at
and the diameter including
intersect the circle again at
.
Use power of a point on point C to the circle centered at A.
So
.
Obviously so we have three solution pairs for
.
By the Triangle Inequality, only
yields a possible length of
.
Therefore, the answer is .
Solution 4
We first draw the height of isosceles triangle and get two equations by the Pythagorean Theorem.
First,
. Second,
.
Subtracting these two equations, we get
.
We then add
to both sides to get
.
We then complete the square to get
. Because
and
are both integers, we get that
is a square number. Simple guess and check reveals that
.
Because
equals
, therefore
. We want
, so we get that
.
See Also
2013 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.