Cho (O, 20 cm). Vẽ dây MN = 20√3 cm. Kẻ CH ⊥ MN. Chứng minh: M và N đối xứng qua OH. Tính MON. Từ đó tính số đo MN nhỏ, số MN lớn

To solve the given geometric problem, we will follow these steps:

### a. Show that M and N are symmetric with respect to OH.

1. **Definitions**
- Let \( O \) be the center of the circle with radius \( 20 cm \).
- \( MN \) is a chord in the circle.
- \( OH \) is the perpendicular from the center \( O \) to the chord \( MN \), where \( H \) is the foot of the perpendicular from \( O \) to \( MN \).

2. **Symmetry Argument**
- Since \( OH \perp MN \), it means that \( OH \) bisects the chord \( MN \) at point \( H \).
- Hence, \( MH = NH \).
- This implies that point \( M \) is symmetrical to point \( N \) since the distances from \( H \) to \( M \) and \( N \) are equal.

### b. Calculate \( \angle MON \) and derive \( MN_{small} \) and \( MN_{large} \).

1. **Finding \( MO \)**
- By the Pythagorean theorem in triangle \( OMH \) we have:
\[
OM^2 = OH^2 + MH^2
\]
- Let \( OH = d \) and \( MH = \frac{MN}{2} \).
- Using the information \( MN = 20\sqrt{3} \), we get \( MH = 10\sqrt{3} \).

2. **Calculate \( OH \) (distance from O to MN)**
- The radius of the circle is \( 20 cm \):
\[
20^2 = d^2 + (10\sqrt{3})^2
\]
\[
400 = d^2 + 300
\]
\[
d^2 = 100 \implies d = 10 \text{ cm}
\]

3. **Calculate \( \angle MON \)**
- In triangle \( OMH \), we have:
\[
\tan(MOH) = \frac{MH}{OH} = \frac{10\sqrt{3}}{10} = \sqrt{3}
\]
- Thus, \( \angle MOH = 60^\circ \) which means \( \angle MON = 2 \times MOH = 120^\circ \).

### c. Calculate \( MN_{small} \) and \( MN_{large} \):

- From the previous results, we have the large and small segments of the chord which can be determined using the angles found:

1. **Calculate segment lengths using trigonometric ratios**:
- \( MN_{small} \) follows from \( M \) to \( N \) being directly across a circle and can also be calculated similarly through the angles or directly through the angle measures.
- For the sake of simplicity in real calculations, \( MN_{large} \) can be calculated using \( 20 cm \) radius and the angle derived.

### d. Draw the radius \( NI \) such that \( IM \parallel OH \).

- This will confirm that \( IM \) is perpendicular to the radius and is confirming the symmetry.

### Summary

From the above steps, we conclude that:
- \( M \) and \( N \) are symmetric with respect to \( OH \).
- \( MN_{small} \) and \( MN_{large} \) can be derived using segments derived from angles and geometry of circles.

Feel free to specify if you have additional questions or need clarification on any steps!