In a `Delta ABC`, if `AB + BC = 10 cm, BC + CA = 12 cm and CA + AB = 16 cm`, then the perimeter of the triangle is _____

Options

A. 19cm
B. 17cm
C. 28cm
D. 22cm

Answer

To find the perimeter of triangle ABC given the equations:

1. \( AB + BC = 10 \, \text{cm} \)
2. \( BC + CA = 12 \, \text{cm} \)
3. \( CA + AB = 16 \, \text{cm} \)

we can follow these steps:

### Step 1: Add all three equations
We start by adding the three equations together:

\[ (AB + BC) + (BC + CA) + (CA + AB) = 10 + 12 + 16 \]

### Step 2: Simplify the left side
On the left side, we can combine like terms:

\[ 2AB + 2BC + 2CA = 38 \]

### Step 3: Factor out the 2
Now, we can factor out the 2 from the left side:

\[ 2(AB + BC + CA) = 38 \]

### Step 4: Divide by 2
To find \( AB + BC + CA \), we divide both sides by 2:

\[ AB + BC + CA = \frac{38}{2} = 19 \, \text{cm} \]

### Step 5: Conclusion
The perimeter of triangle ABC is the sum of its sides:

\[ \text{Perimeter} = AB + BC + CA = 19 \, \text{cm} \]

Thus, the perimeter of triangle ABC is **19 cm**.

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