Q. 124.1( 57 Votes )

# 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 T_{1} with sides 3, 4, 5, T_{2} with sides 5, T_{2}, 13 and T_{3} with sides 6, 8, 10. Which triangles among T_{1}, T_{2} and T_{3} are related?

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}.

Rate this question :

Fill in the blanks in each of the

Let the relation R be defined on the set

A = {1, 2, 3, 4, 5} by R = {(a, b) : |a^{2} – b^{2}| < 8}. Then R is given by _______.

State True or False for the statements

Every relation which is symmetric and transitive is also reflexive.

Mathematics - ExemplarState True or False for the statements

Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.

Mathematics - ExemplarState True or False for the statements

An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.

Mathematics - Exemplar