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The image presents a geometry problem involving a triangle \( ABC \), where the sides \( AB = 4 \), \( AC = 6 \), and the angle \( \hat{A} = 30^\circ \). We are tasked with finding the height \( CH \), which is perpendicular to \( AB \) from point \( C \).

### Step-by-step approach:

1. **Using the area formula**:
   The area of triangle \( ABC \) can be found in two ways:
   
   - **Using the sine rule**:
     The formula for the area of a triangle is:
     \[
     \text{Area} = \frac{1}{2} \times AB \times AC \times \sin(\hat{A})
     \]
     Substituting the known values:
     \[
     \text{Area} = \frac{1}{2} \times 4 \times 6 \times \sin(30^\circ)
     \]
     Since \( \sin(30^\circ) = \frac{1}{2} \):
     \[
     \text{Area} = \frac{1}{2} \times 4 \times 6 \times \frac{1}{2} = 6
     \]

2. **Using the height \( CH \)**:
   The area can also be written as:
   \[
   \text{Area} = \frac{1}{2} \times AB \times CH
   \]
   Since the area is 6, we substitute:
   \[
   6 = \frac{1}{2} \times 4 \times CH
   \]
   Solving for \( CH \):
   \[
   CH = \frac{6 \times 2}{4} = 3
   \]

Thus, the height \( CH \) is \( 3 \).

### Answer:
The correct answer is option \( 1 \), which is \( 3 \).

Do you have any further questions, or would you like a more detailed explanation?

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Here are 5 related questions you might find interesting:

1. How is the area of a triangle calculated using the sine of an angle?
2. What is the relationship between the base and height of a triangle in calculating its area?
3. Can the height of a triangle always be drawn inside the triangle?
4. How do you apply trigonometric functions to find unknown sides or heights in non-right triangles?
5. What other methods can be used to find the height of a triangle?

**Tip**: Always remember that the sine function for angles of 30°, 45°, and 60° has specific values that are useful for solving geometric problems quickly!

In triangle ABC, AB = 4, AC = 6, and angle A = 30 degrees. What is the length of height CH?

Find the height of triangle ABC where AB = 4, AC = 6, and angle A = 30 degrees using trigonometric and geometric principles. A detailed step-by-step solution involves using the sine rule and the area formula.

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