If the areas of two similar triangles are equal, prove that they are congruent
As we know if three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
Let's consider two similar triangles ΔABC and ΔDEF
Thus, ΔABC ∼ ΔDEF
⇒ AB/DE = BC/EF = CA/FD (SSS criterion)
But, Area of ΔABC = Area of ΔDEF (According to the given question)
⇒ Area of ΔABC / Area of ΔDEF = 1 ................. (1)
Area of ΔABC / Area of ΔDEF = (AB)2 / (DE)2 = (BC)2 / (EF)2 = (CA)2 / (FD)2 (According to theorem 6.6) ............... (2)
From equation (1) and (2),
(AB)2 / (DE)2 = 1
⇒ (AB)2 = (DE)2
⇒ AB = DE ............... (3)
⇒ BC = EF ....(4)
⇒ CA = FD .....(5)
In ΔABC and ΔDEF
⇒ AB = DE [from equation(3)]
⇒ BC = EF [from equation (4)]
⇒ CA = FD [from equation (5)]
Thus, ΔABC ≅ ΔDEF (SSS congruency)
If the areas of two similar triangles are equal, prove that they are congruent.
NCERT Class 10 Maths Solutions Chapter 6 Exercise 6.4 Question 4
If the areas of two similar triangles are equal, we have proved that they are congruent.
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