To solve the problem, we need to analyze the given information about triangle DEF, which is divided into two triangles, DEM and DFM, of equal areas. We will evaluate each option to determine which statements are true.
### Step-by-Step Solution:
1. **Understanding the Area of Triangles**:
- The area of a triangle can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
- Since triangles DEM and DFM have equal areas, we can express this mathematically:
\[
\text{Area of } \triangle DEM = \text{Area of } \triangle DFM
\]
2. **Analyzing Option A**:
- Option A states that triangles DEF and DFM have equal bases.
- If we denote the base of triangle DEM as EM and the base of triangle DFM as FM, we can set up the equation:
\[
\frac{1}{2} \times EM \times h = \frac{1}{2} \times FM \times h
\]
- Since the height (h) is the same for both triangles, we can cancel it out, leading to:
\[
EM = FM
\]
- This means that the bases of triangles DEM and DFM are equal. Therefore, **Option A is true**.
3. **Analyzing Option B**:
- Option B claims that the area of triangle DEM is equal to \(\frac{1}{3}\) of the area of triangle DEF.
- Since we know that the areas of triangles DEM and DFM are equal, we can express the area of triangle DEF as:
\[
\text{Area of } \triangle DEF = \text{Area of } \triangle DEM + \text{Area of } \triangle DFM = 2 \times \text{Area of } \triangle DEM
\]
- Therefore, the area of triangle DEM is actually \(\frac{1}{2}\) of the area of triangle DEF, not \(\frac{1}{3}\). Thus, **Option B is false**.
4. **Analyzing Option C**:
- Option C states that M is the midpoint of side EF.
- Since we established that EM = FM, point M must indeed be the midpoint of side EF. Therefore, **Option C is true**.
5. **Analyzing Option D**:
- Option D claims that triangles DEM and DFM are congruent.
- To check for congruence, we can use the SSS (Side-Side-Side) criterion. We have:
- EM = FM (from Option A)
- DM is common to both triangles.
- However, we do not have information about the lengths DE and DF, so we cannot conclude that the triangles are congruent. Therefore, **Option D is false**.
### Conclusion:
The true statements are:
- **Option A**: DEM and DFM have equal bases.
- **Option C**: M is the midpoint of side EF.
