The median of a triangle divides it into two

Options

A. triangles of equal area
B. congruent triangles
C. right angled triangles
D. isosceles triangles

Explanation

To solve the problem, we need to demonstrate that the median of a triangle divides it into two triangles of equal area. Let's break this down step by step.

### Step-by-Step Solution:

1. **Understanding the Triangle and Median**:
- Consider a triangle \( ABC \).
- Let \( D \) be the midpoint of side \( BC \).
- The line segment \( AD \) is the median of triangle \( ABC \).

**Hint**: Remember that a median connects a vertex to the midpoint of the opposite side.

2. **Identifying the Triangles**:
- The median \( AD \) divides triangle \( ABC \) into two smaller triangles: \( ABD \) and \( ACD \).

**Hint**: Visualize the triangle and the median to see how it splits the area.

3. **Calculating the Area of Triangles**:
- The area of triangle \( ABD \) can be calculated using the formula:
\[ \text{Area}_{ABD} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \( BD \) and the height is the perpendicular distance from \( A \) to line \( BC \).

**Hint**: The height is the same for both triangles since they share the vertex \( A \) and the line \( BC \) as the base.

4. **Area of Triangle \( ACD \)**:
- Similarly, the area of triangle \( ACD \) is:
\[ \text{Area}_{ACD} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \( CD \) and the height remains the same as for triangle \( ABD \).

**Hint**: Identify that both triangles share the same height from vertex \( A \) to line \( BC \).

5. **Equal Lengths of Bases**:
- Since \( D \) is the midpoint of \( BC \), we have:
\[ BD = CD \]
- Therefore, the bases of triangles \( ABD \) and \( ACD \) are equal.

**Hint**: Use the property of midpoints to establish that \( BD \) and \( CD \) are equal.

6. **Setting Up the Area Equation**:
- Now, substituting the equal lengths of bases into the area formulas:
\[ \text{Area}_{ABD} = \frac{1}{2} \times BD \times h \]
\[ \text{Area}_{ACD} = \frac{1}{2} \times CD \times h \]
- Since \( BD = CD \), we can say:
\[ \text{Area}_{ABD} = \text{Area}_{ACD} \]

**Hint**: This shows that both triangles have the same area due to equal bases and the same height.

7. **Conclusion**:
- Therefore, we conclude that the median \( AD \) divides triangle \( ABC \) into two triangles \( ABD \) and \( ACD \) that have equal areas.

**Hint**: This confirms that the median of a triangle divides it into two triangles of equal area.

### Final Answer:
The median of a triangle divides it into two triangles of equal area.