Q.11 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 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?

$R = \{(T_1 , T_2 ) : T_1 \;is\; similar \;to\; T_2 \}$

All triangles are similar to itself, so it is reflexive.

Let,

$(T_1,T_2) \in R$  i.e.T1 is similar to T2

T1 is similar to T2 is same asT2 is similar to T1 i.e. $(T_2,T_1) \in R$

Hence,it is symmetric.

Let,

$(T_1,T_2) \in R$  and  $(T_2,T_3) \in R$  i.e. T1 is similar to T2  and T2 is similar toT3 .

$\Rightarrow$T1 is similar toT3   i.e. $(T_1,T_3) \in R$

Hence, it is transitive,

Thus,  $R = \{(T_1 , T_2 ) : T_1 \;is\; similar \;to\; T_2 \}$, is equivalence relation.

Now , we see ratio of sides of triangle T1 andT3 are as shown

$\frac{3}{6}=\frac{4}{8}=\frac{5}{10}=\frac{1}{2}$

i.e. ratios of sides of T1 and T3 are equal.Hence, T1 and T3 are related.

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