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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:

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:

\begin{align} \frac {3}{6}=\frac {4}{8}=\frac {5}{10} = \left(\frac {1}{2}\right) \end{align}

∴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.

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