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 T₂.
⇒ T₂ is similar to T₁. 
⇒ (T₂, T₁) ∈R 
∴R is symmetric

Now,
Let (T₁, T₂), (T₂, T₃) ∈ R.
⇒ T₁ is similar to T₂ and T₂ is similar to T₃.
⇒ T₁ is similar to T₃.
⇒ (T₁, T₃) ∈ R
∴ R is transitive.

Thus, R is an equivalence relation.
Now, we can observe that:
\(\frac {3} {6}\)=\(\frac {4} {8}\)= \(\frac {5} {10}\)= \(\bigg(\frac{1}{2}\bigg)\)
∴The corresponding sides of triangles T₁ and T₃ are in the same ratio.
Then, triangle T₁ is similar to triangle T₃.

Hence, T₁ is related to T₃.