In a `DeltaABC,D,E and F` are respectively the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are 7 cm, 8 cm and 9 cm respectively, find the permeter of `DeltaDEF.`

Answer

To find the perimeter of triangle DEF, where D, E, and F are the midpoints of sides BC, CA, and AB of triangle ABC respectively, we can follow these steps:

### Step 1: Identify the lengths of the sides of triangle ABC.
- Given:
- AB = 7 cm
- BC = 8 cm
- CA = 9 cm

### Step 2: Use the Midpoint Theorem to find the lengths of the sides of triangle DEF.
According to the Midpoint Theorem, the length of a segment connecting the midpoints of two sides of a triangle is half the length of the third side.

#### Step 2.1: Calculate the length of side EF.
- EF is the segment connecting midpoints E and F, which are midpoints of sides CA and AB respectively.
- Using the Midpoint Theorem:
\[ EF = \frac{1}{2} \times BC = \frac{1}{2} \times 8 = 4 \text{ cm} \]

#### Step 2.2: Calculate the length of side DE.
- DE is the segment connecting midpoints D and E, which are midpoints of sides BC and CA respectively.
- Using the Midpoint Theorem:
\[ DE = \frac{1}{2} \times AB = \frac{1}{2} \times 7 = 3.5 \text{ cm} \]

#### Step 2.3: Calculate the length of side DF.
- DF is the segment connecting midpoints D and F, which are midpoints of sides BC and AB respectively.
- Using the Midpoint Theorem:
\[ DF = \frac{1}{2} \times CA = \frac{1}{2} \times 9 = 4.5 \text{ cm} \]

### Step 3: Calculate the perimeter of triangle DEF.
- The perimeter of triangle DEF is the sum of the lengths of its sides:
\[ \text{Perimeter} = EF + DE + DF \]
Substituting the values we found:
\[ \text{Perimeter} = 4 + 3.5 + 4.5 \]
Calculating the sum:
\[ \text{Perimeter} = 12 \text{ cm} \]

### Final Answer:
The perimeter of triangle DEF is **12 cm**.

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