Question

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, 12, 13 and T 3 with sides 6, 8, 10. Which triangles among T 1 , T 2 and T 3 are related?

Answer 1

The given relation R in the set A of all triangles is defined as R={ ( T 1 , T 2 ): T 1  is similar to  T 2 }.

Every triangle is similar to itself. So, R is reflexive.

Let, ( T 1 , T 2 )∈R

⇒ T 1  is similar to  T 2 ⇒ T 2  is similar to  T 1 ⇒( T 2 , T 1 )∈R

So, R is symmetric.

Let, ( T 1 , T 2 )∈R and ( T 2 , T 3 )∈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 )∈R

So, R is transitive.

Thus, R is an equivalence relation.

The given right triangles are T 1 with sides ( 3,4,5 ), T 2 with sides ( 5,12,13 ) and T 3 with sides ( 6,8,10 ).

In triangles T 1 and T 3 , the ratio of sides is,

3 6 = 4 8 = 5 10 = 1 2

Since the ratio of sides is equal, hence, triangles T 1 and T 3 are similar, so they are related to each other.

T 2 is not similar to T 1 and T 3 .

Thus, T 2 is not related to either T 1 or T 3 .