1. **State the problem:** We have a circle with center $O$ and two chords inside it. We need to find the lengths $a$ and $b$ of parts of these chords. 2. **Recall the property of chords and radius:** The radius drawn perpendicular to a chord bisects the chord. This means the radius from $O$ to the chord divides the chord into two equal parts. 3. **Find length $a$:** Given one chord is divided into segments $18$ cm and $16$ cm, and the radius $O$ to this chord is perpendicular at the $16$ cm segment. Since the radius bisects the chord, the two segments must be equal: $$a = 18 \text{ cm}$$ 4. **Find length $b$:** For the other chord, the radius $O$ is perpendicular to the chord segment labeled $b$, and the other segment is $24$ cm. Since the radius bisects the chord: $$b = 24 \text{ cm}$$ 5. **Reasoning:** The key reason is that the radius perpendicular to a chord bisects the chord, so the two parts of each chord are equal. **Final answers:** $$a = 18 \text{ cm}, \quad b = 24 \text{ cm}$$