M and N are points on sides AC and BC respectively of a `triangleABC`. State whether MN||BA` if CM = 4.2 cm, MA= 2.8 cm , NB = 3.6 cm, CN = 5.7 cm.

Answer

To determine whether the line segment MN is parallel to the line segment AB in triangle ABC, we can use the concept of ratios derived from the Thales theorem. Here’s a step-by-step solution:

### Step 1: Identify the Given Lengths
We are given the following lengths:
- CM = 4.2 cm
- MA = 2.8 cm
- CN = 5.7 cm
- NB = 3.6 cm

### Step 2: Apply the Reverse Thales Theorem
According to the reverse Thales theorem, if a line segment is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. Therefore, we need to check if:

\[ \frac{CM}{MA} = \frac{CN}{NB} \]

### Step 3: Calculate the Ratios
First, we calculate the ratio \(\frac{CM}{MA}\):

\[ \frac{CM}{MA} = \frac{4.2 \text{ cm}}{2.8 \text{ cm}} = 1.5 \]

Next, we calculate the ratio \(\frac{CN}{NB}\):

\[ \frac{CN}{NB} = \frac{5.7 \text{ cm}}{3.6 \text{ cm}} \approx 1.5833 \]

### Step 4: Compare the Ratios
Now we compare the two ratios we calculated:

- \(\frac{CM}{MA} = 1.5\)
- \(\frac{CN}{NB} \approx 1.5833\)

Since \(1.5 \neq 1.5833\), the ratios are not equal.

### Step 5: Conclusion
Since the ratios are not equal, we conclude that the line segment MN is **not parallel** to the line segment AB.

### Final Answer
Therefore, MN is not parallel to AB.

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