Q. 124.0( 68 Votes )

# Show that the rel

Answer :

It is given 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}},

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

Now, if (T_{1}, T_{2}) ϵ R, then T_{1} is similar to T_{2}.

⇒ T_{2} is similar to T_{1}.

⇒ (T_{1}, T_{2}) ϵ R

Therefore, R is symmetric.

Now, if (T_{1}, T_{2}), (T_{2}, T_{3}) ϵ R,

⇒ T_{1} is similar to T_{2} and T_{2} is similar to T_{3}.

⇒ T_{1} is similar to T_{3}.

⇒ (T_{1}, T_{3}) ϵ R

Therefore, R is transitive.

Therefore, R is equivalence relation.

Now, we can see that,

Therefore, the corresponding sides of triangles T_{1} and T_{3} are in the same ratio.

Thus, triangle T_{1} is similar to triangle T_{3}.

Therefore, T_{1} is related to T_{3}.

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