if the areas of two similar triangles are equal prove that they are congruent - Maths -

Question

if the areas of two similar triangles are equal prove that they
are congruent

Ankush Jain · Accepted Answer

Consider two triangles viz , Î”ABC and Î”DEF which are similar to each other Given, Area of Î”ABC = Area of Î”DEF Now, we know that ratio of area of two similar triangle is equal to the ratio of squares of their corresponding sides $â‡’ \frac{Area of ï¿½ Î” ABC}{Area of ï¿½ Î” DEF} = (\frac{AB}{DE})^{2} = (\frac{BC}{EF})^{2} = (\frac{AC}{DF})^{2} â‡’ 1 = (\frac{AB}{DE})^{2} = (\frac{BC}{EF})^{2} = (\frac{AC}{DF})^{2} â‡’ 1 = \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} â‡’ \frac{AB}{DE} = 1, \frac{BC}{EF} = 1, \frac{AC}{DF} = 1 â‡’ AB = DE, â€‰ â€‰ â€‰ BC = EF â€‰ â€‰ â€‰ â€‰ â€‰ â€‰ â€‰ and â¢ â€‰ â€‰ â€‰ â€‰ â€‰ AC = DF$ âˆ´ All the sides of Î”ABC are equal to the corresponding sides of Î”DEF $Hence, ï¿½ï¿½ Î” ABC â‰\dots Î” DEF$ [SSS property]

if the areas of two similar triangles are equal prove that they are congruent .