Show that the relation R defined in the set A of all triangles as R = {(T 1 , T 2 ): T 1 is similar to T 2 }, is equivalence relation. Consider three right angle triangles T 1 with sides 3, 4, 5, T 2 with sides 5, T 2 , 13 and T 3 with sides 6, 8, 10. Which triangles among T 1 , T 2 and T 3 are related?

Question

Philoid · Accepted Answer

It is given that the relation R defined in the set A of all triangles as

R = {(T₁, T₂): T₁ is similar to T₂},

Now, R is reflexive as every triangle is similar to itself.

Now, if (T₁, T₂) ϵ R, then T₁ is similar to T₂.

⇒ T₂ is similar to T₁.

⇒ (T₁, T₂) ϵ R

Therefore, R is symmetric.

Now, if (T₁, T₂), (T₂, T₃) ϵ R,

⇒ T₁ is similar to T₂ and T₂ is similar to T₃.

⇒ T₁ is similar to T₃.

⇒ (T₁, T₃) ϵ R

Therefore, R is transitive.

Therefore, R is equivalence relation.

Now, we can see that,

Therefore, the corresponding sides of triangles T₁ and T₃ are in the same ratio.

Thus, triangle T₁ is similar to triangle T₃.

Therefore, T₁ is related to T₃.

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, T2, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?