The difference between the semi-perimeter and the sides of a `DeltaABC` are 7 cm, 5 cm and 3 cm respectively. The perimeter of the triangle is____

Options

A. 25cm
B. 10cm
C. 15cm
D. 30cm

Answer

To find the perimeter of triangle ABC given the differences between the semi-perimeter and the sides, we can follow these steps:

### Step 1: Define the semi-perimeter and sides
Let the semi-perimeter of the triangle be denoted as \( S \). The sides of the triangle are denoted as \( A \), \( B \), and \( C \). According to the problem, we have the following equations based on the differences given:

1. \( S - A = 7 \) (Equation 1)
2. \( S - B = 5 \) (Equation 2)
3. \( S - C = 3 \) (Equation 3)

### Step 2: Rearrange the equations
From the equations above, we can express the sides in terms of the semi-perimeter \( S \):

- From Equation 1: \( A = S - 7 \)
- From Equation 2: \( B = S - 5 \)
- From Equation 3: \( C = S - 3 \)

### Step 3: Add the equations
Now, we add all three equations together:

\[ (S - A) + (S - B) + (S - C) = 7 + 5 + 3 \]

This simplifies to:

\[ 3S - (A + B + C) = 15 \]

### Step 4: Express the perimeter
We know that the perimeter \( P \) of the triangle is given by:

\[ P = A + B + C \]

Substituting \( A + B + C \) from the previous step, we can rewrite the equation:

\[ A + B + C = 3S - 15 \]

### Step 5: Relate semi-perimeter to perimeter
Since the semi-perimeter \( S \) is half of the perimeter, we have:

\[ S = \frac{P}{2} \]

Substituting this into the equation gives:

\[ A + B + C = 3 \left(\frac{P}{2}\right) - 15 \]

This simplifies to:

\[ A + B + C = \frac{3P}{2} - 15 \]

### Step 6: Set the equations equal
Since we know \( A + B + C = P \), we can set the two expressions for \( A + B + C \) equal to each other:

\[ P = \frac{3P}{2} - 15 \]

### Step 7: Solve for \( P \)
To solve for \( P \), we can rearrange the equation:

\[ P - \frac{3P}{2} = -15 \]

This simplifies to:

\[ -\frac{P}{2} = -15 \]

Multiplying both sides by -2 gives:

\[ P = 30 \]

### Conclusion
The perimeter of triangle ABC is \( \boxed{30 \text{ cm}} \).

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