In triangle ABC, D and E are mid-points of sides AB and BC respectively. Also, F is a point in side AC so that DF is parallel to BC
Find the perimeter of parallelogram DBEF, if AB = 10 cm, BC = 8.4 cm and AC = 12 cm.

Answer

To find the perimeter of the parallelogram DBEF in triangle ABC, we will use the properties of midpoints and parallel lines. Here’s a step-by-step solution:

### Step 1: Identify the Midpoints
In triangle ABC:
- D is the midpoint of side AB.
- E is the midpoint of side BC.
- F is a point on side AC such that DF is parallel to BC.

### Step 2: Apply the Midpoint Theorem
According to the midpoint theorem:
- Since D is the midpoint of AB and E is the midpoint of BC, the line segment DE is parallel to side AC and DE = 1/2 * AC.
- Since DF is parallel to BC and D is the midpoint of AB, by the converse of the midpoint theorem, F must also be the midpoint of AC.

### Step 3: Calculate Lengths of Segments
- Given:
- AB = 10 cm
- BC = 8.4 cm
- AC = 12 cm

Since D is the midpoint of AB:
- DB = 1/2 * AB = 1/2 * 10 cm = 5 cm

Since E is the midpoint of BC:
- BE = 1/2 * BC = 1/2 * 8.4 cm = 4.2 cm

Since F is the midpoint of AC:
- AF = 1/2 * AC = 1/2 * 12 cm = 6 cm

### Step 4: Find Lengths of DF and EF
Since DF is parallel to BC:
- DF = 1/2 * BC = 1/2 * 8.4 cm = 4.2 cm

Since FE is parallel to AB:
- FE = 1/2 * AB = 1/2 * 10 cm = 5 cm

### Step 5: Calculate the Perimeter of Parallelogram DBEF
The perimeter of parallelogram DBEF is given by:
\[ \text{Perimeter} = DB + BE + EF + DF \]

Since DB = EF and BE = DF, we can simplify:
\[ \text{Perimeter} = 2(DB + BE) \]
\[ \text{Perimeter} = 2(5 cm + 4.2 cm) \]
\[ \text{Perimeter} = 2(9.2 cm) \]
\[ \text{Perimeter} = 18.4 cm \]

### Final Answer
The perimeter of parallelogram DBEF is **18.4 cm**.

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