1. **Problem Statement:** In circle $\odot O$, chord $AB$ and segment $OC$ intersect at point $M$. Given that $OM = MC$ and $AM = 1.5$, $BM = 4$, find the length of $OC$. 2. **Relevant Theorem:** When two chords intersect inside a circle, the products of the segments of each chord are equal. This is called the Intersecting Chords Theorem. 3. **Formula:** If chords $AB$ and $CD$ intersect at $M$, then: $$AM \times MB = CM \times MD$$ 4. **Apply to our problem:** Here, $AB$ is a chord with segments $AM = 1.5$ and $BM = 4$. The segment $OC$ is divided into two equal parts by $M$, so $OM = MC = x$ (unknown). 5. Using the theorem: $$AM \times BM = OM \times MC$$ $$1.5 \times 4 = x \times x$$ $$6 = x^2$$ 6. Solve for $x$: $$x = \sqrt{6}$$ 7. Since $OC = OM + MC = x + x = 2x$, then: $$OC = 2 \times \sqrt{6} = 2\sqrt{6}$$ **Final answer:** $$OC = 2\sqrt{6}$$