If D, E and F are respectively the mid - points of sides BC, AC and AB of a `DeltaABC`. If `EF = 3cm, FD=4cm` and `AB=10 cm,` then `DE, BC and CA` respectively will be equal to :

Options

A. 6, 8 and 20 cm
B. 4, 6 and 8 cm
C. 5, 6 and 8 cm
D. `(10)/(3)`, 9 and 12 cm

Answer

To solve the problem step by step, we will use the Midpoint Theorem and properties of similar triangles.

### Step 1: Understand the given information
We have triangle ABC with midpoints D, E, and F on sides BC, AC, and AB respectively. We are given:
- EF = 3 cm
- FD = 4 cm
- AB = 10 cm

### Step 2: Use the Midpoint Theorem
According to the Midpoint Theorem, if D, E, and F are midpoints of sides BC, AC, and AB respectively, then:
- DE is parallel to AB
- EF is parallel to AC
- FD is parallel to BC

### Step 3: Find DE
Since DE is parallel to AB and D and E are midpoints, we can use the ratio of the sides:
- Since AB = 10 cm, and DE is half of AB, we can calculate DE as follows:

\[ DE = \frac{AB}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]

### Step 4: Find BC
Next, we need to find the length of BC. We know that:
- EF is parallel to AC, and since E and F are midpoints, we can use the ratio of the sides again:

Using the triangle AFE and triangle ABC, we have:
\[ \frac{AF}{AB} = \frac{EF}{BC} \]
Since AF is half of AB (because F is the midpoint), we have:
\[ \frac{1}{2} = \frac{3 \text{ cm}}{BC} \]
Cross-multiplying gives:
\[ BC = 3 \text{ cm} \times 2 = 6 \text{ cm} \]

### Step 5: Find CA
Now we need to find CA. We can use triangle BFD and triangle BAC:
\[ \frac{BF}{BA} = \frac{DF}{CA} \]
Since BF is half of BA (because F is the midpoint), we have:
\[ \frac{1}{2} = \frac{4 \text{ cm}}{CA} \]
Cross-multiplying gives:
\[ CA = 4 \text{ cm} \times 2 = 8 \text{ cm} \]

### Final Results
Now we have:
- DE = 5 cm
- BC = 6 cm
- CA = 8 cm

Thus, the lengths of DE, BC, and CA are 5 cm, 6 cm, and 8 cm respectively.

### Summary of Results
The values are:
- DE = 5 cm
- BC = 6 cm
- CA = 8 cm