Show that the relation R defined in the set A of all triangles as R = {(T1 , T2 ) : T1 is similar to T2 }, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1 , T2 and T3 are related?

Solution:

R = {(T₁, T₂) : T₁ is similar to T₂}

R is reflexive since every triangle is similar to itself.

If (T₁, T₂) ∈ R, then T1 is similar to T₂.

T₂ is similar to T₁.

⇒ (T₂, T₁) ∈ R

∴ R is symmetric.

(T₁, T₂), (T₂, T₃) ∈ R

T₁ is similar toT₂ and T₂ is similar to T₃.

∴ T₁ is similar to T₃.

⇒ (T₁, T₃) ∈ R

∴ R is transitive.

3/6 = 4/8 = 5/10 = (1/2)

∴ Corresponding sides of triangles T₁ and T₃ are in the same ratio.

Triangle T₁ is similar to triangle T₃.

Hence, T₁ is related to T₃

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.1 Question 12

Show that the relation R defined in the set A of all triangles as R = {(T1, T2 ) : T1 is similar to T2 }, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?

Summary:

Here we have observed that  T₁ is similar toT₂ and T₂ is similar to T₃ therefore  T₁ is similar to T₃. Hence the given relation is an equivalence relation