+1248 Chords $\overline{AB}$ and $\overline{CD}$ of a circle meet at $P$. If $AP = 3 \cdot PB$, then what is $AB$?
+193 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.
