Question

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

Answer 1

$\text{\bf{equivalence relation}}$:- A relation R is said to be an equivalence relation. if it is simultaneously reflexive, symmetric and transitive on A.

Given, R = {(T1,T2) : T1 is similar to T2}
here, every triangle is similar to itself.
so, R is reflexive.
further , if (T1,T2) ∈ R, then T1 is similar to T2.
=> T2 is similar to T1.
=> (T2, T1)∈R
hence, R is symmetric.

now, Let (T1,T2),(T2,T3)∈R
=> T1 is similar to T2 and TWO is similar to THE.
=> T1 is similar to T3.
=> (T1, T3) ∈ R
hence, R is transitive.

thus , R is an equivalence relation.
now, we can observe that,
$\frac{3}{6}=\frac{4}{8}=\frac{1}{2}$
so, the corresponding sides of triangles T1 and T3 are in the same ratio.
then, triangle T1 is similar to triangle T3.
Hence, T1 is related to T3.