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

[asy]
unitsize(0.4 cm);

pair A, B, T, X, Y;

T = (0,0);
A = 4*sqrt(3)*dir(250);
B = 2*sqrt(3)*dir(70);
X = 4*dir(190);
Y = 6*dir(10);

draw(circumcircle(A,B,X));
draw(A--B);
draw(X--Y);

label("$A$", A, SW);
label("$B$", B, N);
label("$P$", T, SE);
label("$C$", X, W);
label("$D$", Y, E);
label("$6$", (T + X)/2, N, red);
label("$12$", (T + Y)/2, N, red);
[/asy]


We are given that the chords $\overline{AB}$ and $\overline{CD}$ intersect at point $P$, and that $AP = 3 \cdot PB$. Our goal is to find the length of $\overline{AB}$.

Let $AP = x$ and $PB = y$. According to the problem, we are given that:

\[
AP = 3 \cdot PB \quad \text{or} \quad x = 3y.
\]

Thus, the length of $AB$ is the sum of $AP$ and $PB$, which is:

\[
AB = AP + PB = x + y.
\]

Substituting $x = 3y$, we get:

\[
AB = 3y + y = 4y.
\]

Therefore, the length of $AB$ is $4y$, where $y = PB$. This expression represents the length of chord $AB$, but since no additional information (such as the lengths of the other segments) is provided, the final answer is $AB = 4y$.

Let me know if you need more details or any further questions. Here are some related queries you might consider:

1. What is the length of chord $CD$ given other conditions?
2. How can we determine the value of $y$ if more information is provided?
3. What is the geometric significance of the ratio $AP:PB$?
4. How does the Power of a Point theorem relate to this problem?
5. What additional properties of circles are useful for solving chord-related problems?

**Tip:** In circle problems involving intersecting chords, the Power of a Point theorem is often useful to relate the lengths of the segments formed.

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

Explore how to calculate the length of chord AB when AP is 3 times PB. Learn the key concepts of circle geometry and how the Power of a Point theorem applies to intersecting chords. Ideal for Grades 8-10.

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