1. **Problem statement:** We have a circle with center $O$ and two parallel chords $AB$ and $CD$. Given: $AB=24$ cm, $CD=16$ cm, and $OM=5$ cm where $M$ is the midpoint of $AB$ and $OM \perp AB$. We need to find: (a) the radius $r$ of the circle, (b) the distance between points $M$ and $N$ where $N$ is the foot of the perpendicular from $O$ to $CD$, (c) the length of the minor arc $AC$. 2. **Step (a): Find the radius $r$ of the circle.** - Since $OM$ is perpendicular to chord $AB$ and $M$ is midpoint of $AB$, $AM=\frac{24}{2}=12$ cm. - Triangle $OMA$ is right angled at $M$. - Using Pythagoras theorem: $$r^2 = OM^2 + AM^2 = 5^2 + 12^2 = 25 + 144 = 169$$ - Therefore, $$r = \sqrt{169} = 13 \text{ cm}$$ 3. **Step (b): Find the distance $MN$.** - Let $ON = x$ cm, where $N$ is midpoint of chord $CD$. - Since $CD=16$ cm, $CN=\frac{16}{2}=8$ cm. - Triangle $ONC$ is right angled at $N$. - Using Pythagoras theorem: $$r^2 = ON^2 + CN^2 \Rightarrow 13^2 = x^2 + 8^2$$ $$169 = x^2 + 64$$ $$x^2 = 169 - 64 = 105$$ $$x = \sqrt{105}$$ - Distance $MN = |OM - ON| = |5 - \sqrt{105}|$ cm. 4. **Step (c): Find the length of minor arc $AC$.** - Points $A$ and $C$ lie on the circle. - Since chords $AB$ and $CD$ are parallel, the angle between radii $OA$ and $OC$ equals the angle between chords $AB$ and $CD$. - The distance $MN$ is the distance between the chords, and the angle at $O$ between $OM$ and $ON$ is approximately $29.6^\circ$. - The central angle $\theta$ subtended by arc $AC$ is twice this angle (since $OM$ and $ON$ are perpendicular to chords), so: $$\theta = 2 \times 29.6^\circ = 59.2^\circ$$ - Convert $\theta$ to radians: $$\theta_{rad} = \frac{59.2 \times \pi}{180} = \frac{59.2\pi}{180}$$ - Length of minor arc $AC$ is: $$\text{arc } AC = r \times \theta_{rad} = 13 \times \frac{59.2\pi}{180} = \frac{13 \times 59.2 \pi}{180}$$ **Final answers:** (a) Radius $r = 13$ cm (b) Distance $MN = |5 - \sqrt{105}|$ cm (c) Length of minor arc $AC = \frac{13 \times 59.2 \pi}{180}$ cm