If a , b , c are the sides of a triangle ABC and p is the perimeter of the triangle, then what is vmatrix p+c & a & b c & p+a & b c & a & p+b vmatrix equal to?

Step-by-step solution

Let = vmatrix p+c & a & b c & p+a & b c & a & p+b vmatrix Applying the column operation C₁ C₁ + C₂ + C₃ : = vmatrix p+a+b+c & a & b p+a+b+c & p+a & b p+a+b+c & a & p+b vmatrix Since p = a+b+c is the perimeter of the triangle, we have p+a+b+c = 2p . Substituting this into the determinant: = vmatrix 2p & a & b 2p & p+a & b 2p & a & p+b vmatrix Taking 2p common from the first column C₁ : = 2p vmatrix 1 & a & b 1 & p+a & b 1 & a & p+b vmatrix Applying the row operations R₂ R₂ - R₁ and R₃ R₃ - R₁ : = 2p vmatrix 1 & a &