Difference between revisions of "Power of a Point Theorem/Introductory Problem 3"
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Revision as of 09:27, 1 July 2006
Problem
(ARML) In a circle, chords and
intersect at
. If
and
, find the ratio
Solution
Letting makes
. Similarly, letting
makes
. Thus
and
. We therefore seek
.
From the Power of a Point Theorem we have that
![$x\cdot 4x = 4y\cdot 9y\Rightarrow \left(\frac xy\right)^2 = 9$](http://latex.artofproblemsolving.com/2/e/2/2e24729bce4b1c941a9b81673d9030967e70509b.png)
which gives so we take
.
Finally
![$\frac{5x}{13y}=\frac 5{13}\cdot \frac xy = \frac 5{13}\cdot 3 = \frac{15}{13}.$](http://latex.artofproblemsolving.com/a/b/2/ab2026bc35d70296298df319b62b31aa46c6c8f2.png)
Back to the Power of a Point Theorem.