Question

Show that the relation R, defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is an equivalence relation. Consider three right 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?

Answer 1

(i) In a set of triangles, R = {(T1, T2): T1 is similar to T2},

(a) A triangle T1 is similar to itself. R is reflexive.

(b) If triangle T1 is similar to triangle T2, then T2 is similar to triangle T1. ∴ R is symmetric.

(c) Let T1 is similar to triangle T2 and T2 to T3. Then, triangle T1 is similar to triangle T3. ∴ R is transitive.

Hence, R is an equivalence relation.

(ii) Two triangles are similar if their sides are proportional, blow sides 3, 4, 5 of traiangle T1 are proportional to the sides 6, 8, 10 of triangle T3. Therefore, T1 is related to T3.