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Chords $\overline{AB}$ and $\overline{CD}$ of a circle meet at $P$. If $AP = 3 \cdot PB$, then what is $AB$?

 Mar 20, 2024
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Let's denote lengths:

 

AP=x (since it's a multiple of PB)

 

PB=x/3

 

CP=y (since information about this segment isn't given, we denote it with a variable)

 

PD=y/3 (similar logic as for PB)

 

AB=a (what we're solving for)

 

CD=c (not directly needed, but can be helpful for visualization)

 

Apply the Power of a Point Theorem:

 

The Power of a Point Theorem states that for any point P inside a circle, the product of the lengths of the two segments created by drawing secants from that point to the circle is equal. In our case, point P is inside the circle (since chords intersect within the circle), and we can apply the theorem to both secants AB and CD.

 

For secant AB:

 

AP⋅PB=x⋅3x​=3x2​

 

For secant CD:

 

CP⋅PD=y⋅3y​=3y2​

 

Since both expressions represent the same power of point P, they must be equal:

 

3x2​=3y2​

 

Utilize the given information:

 

We are given that AP=3⋅PB, which translates to x=3⋅3x​ (substituting the values we defined). This simplifies to x=3x​, which implies x=0. However, a chord cannot have zero length. Therefore, our initial assumption (that x represents a positive length) must be incorrect.

 

Here's the correction: We can rewrite the given information as x=3PB=3⋅3x​. Solving for x, we get x2=9. Taking the square root of both sides (remembering positive for lengths), we have x=3.

 

Substitute and solve for AB:

 

Since we found x=3, we can substitute this value back into the equation we obtained from the Power of a Point Theorem:

 

3x2​=3y2​

 

332​=3y2​ (substitute x with 3)

 

3=y2

 

Taking the square root (positive for lengths), we have y=3​.

 

Now, consider segment AP : its total length is x+PB=3+33​=4. Since AP=x=3, segment PB must have a length of PB=14−3​=1.

 

Finally, to find the length of AB, we add the lengths AP and PB:

 

AB=AP+PB=3+1=4​.

 Mar 20, 2024

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