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?

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Vishal Baghel | Contributor-Level 10

9 months ago

The given relation to set A of all triangles is defined as

R= ${(T_{1}, T_{2}) : T_{1}$ is similar to $T_{2}}$

For $T_{1} \in A$ ,

$T_{1}$ is always similar to $T_{1}$

So, $(T_{1}, T_{1}) \in R$ . Hence R is reflexive.

For $T_{1,} T_{2} \in A$ and $(T_{1}, T_{2}) \in R$ we have

$T_{1} \sim T_{2} (s i m i l a r)$

$T_{2} \sim T_{1}$ i.e., $(T_{2}, T_{1}) \in R$

so, R is symmetric.

for, $T_{1}, T_{2}, T_{3} \in A$ and $(T_{1}, T_{2}) \in R$ and $(T_{2}, T_{3}) \in R$

$T_{1} \sim T_{2}$ and $T_{2} \sim T_{3}$

i.e., $T_{1} \sim T_{3}$ $\to (T_{1}, T_{3}) \in R$

so, R is transitive

$∴$ R is an equivalence relation.

Given, sides of $T_{1}$ are 3,4,5

Sides of $T_{2}$ are 5,12,13

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