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In the diagram below, PQ is tangent at P to the circle with center O, point S is inside the circle, and QS intersects the circle at R. If , QR = RS = 3, OS = 2 and PQ = 6, then find the radius of the circle.

 Mar 21, 2020
 #1
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Hey there guest! First, let's start the problem with the knowledge of a (pretty common) theorem in geometry: The power of a point theorem. There are three variations to this theorem, but in this problem, we only need to look at 1 : the variation which involves the tangent of a circle and a secant(a line intersecting two points on the circle. 

 

Firstly, notice we can extend the line QS to intersect the other side of the circle at Z. This is where power of a point comes in handy:

By power of a point:

\(QP^{2} = QR*QZ\)

Substituting known values in:

\(6^{2} = 3 * QZ\)

Dividing by 3, we get:

\(12 = QZ\)

Then RZ = QZ - 3(since that's the length of QR),

RZ = 12 - 3 = 9

Now that we have the length of a chord of Circle O(RZ is a chord), we can use power of a point once again, except another of its variations: with two intersecting chords. 

A handy trick to realize is that the circle's diameter is also a chord, meaning we could find the length with power of a point theorem. 

If we draw a diameter intersecting S, we could make power of a point easier to work with. By power of a point:\((R+2)*(R-2) = RS*SZ = 3*6 = 18\)

where R is the radius of the circle

This simplifies to: 
\(R^{2}-4 = 18\)

\(R^{2} = 22 \)

\(R= \sqrt{22}\)

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 Mar 21, 2020
 #2
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thanks for your help!

Guest Mar 21, 2020
 #3
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No problem!

jfan17  Mar 21, 2020

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