If two triangles are similar & have the same area , then how can we prove they are congruent
The area of a triangle given the length of the three sides as a, b and c is equal to sqrt[s(s - a)(s - b)(s - c)], where s is the semi-perimeter.
Let the similar triangles be PQR and P'Q'R'. As they are similar, the corresponding sides have a common ratio. PQ/P'Q' = QR/Q'R' = RP/R'P' = c . If the length of the sides of P'Q'R' are p, q, r, the length of the sides of PQR is pc, qc, rc
As the triangles are equal in area: sqrt[s(s - p)(s - q)(s - r)] = sqrt[sc(sc - pc)(sc - qc)(sc - rc)]
=> sqrt[s(s - p)(s - q)(s - r)] = c*sqrt[s(s - p)(s - q)(s - r)]
=> c = 1
This gives the length of the corresponding sides of the triangles as the same.
The triangles are proved congruent by the SSS condition.