Find the length of MF.

Solution:

In the given circle, we have two intersecting chords, PA and FB. The intersection point is M.

According to the intersecting chords theorem, when two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord.

For chord PA, the segments are PM and MA. We are given that PM = 9 and MA = 3.

For chord FB, the segments are FM and MB. We are given that MB = 2. Let FM = x.

Applying the intersecting chords theorem:

\( PM \cdot MA = FM \cdot MB \)

Substitute the given values:

\( 9 \cdot 3 = x \cdot 2 \)

\( 27 = 2x \)

Solve for x:

\[ x = \frac{27}{2} \]

\[ x = 13.5 \]

Therefore, the length of MF is 13.5.

Ответ: MF = 13.5