Question

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?

Answer 1

R = {(T₁, T₂): T₁ is similar to T₂}

R is reflexive since every triangle is similar to itself.

Further, if (T₁, T₂) ∈ R, then T₁ is similar to T_2.

⇒ T₂ is similar to T_1.

⇒ (T₂, T₁) ∈R

∴R is symmetric.

Now,

Let (T₁, T₂), (T₂, T₃) ∈ R.

⇒ T₁ is similar to T₂ and T₂ is similar to T_3.

⇒ T₁ is similar to T_3.

⇒ (T₁, T₃) ∈ R

∴ R is transitive.

Thus, R is an equivalence relation.

Now, we can observe that:

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

Then, triangle T₁ is similar to triangle T_3.

Hence, T₁­ is related to T₃.