from a handpicked tutor in LIVE 1-to-1 classes
If the areas of two similar triangles are equal, prove that they are congruent
We know that two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
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)
☛ Check: NCERT Solutions Class 10 Maths Chapter 6
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.
☛ Related Questions:
- D, E and F are respectively the mid-points of sides AB, BC and CA of ∆ ABC. Find the ratio of the areas of ∆ DEF and ∆ ABC
- Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
- Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.
- Tick the correct answer and justify:ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is(A) 2: 1(B) 1: 2(C) 4: 1(D) 1: 4