Question

Show that the relation R defined in the set A of all triangles as $R=\left\{\right(T_1,T_2):T_{1}issimilarto{T}_2\}$, is equivalence relation. Consider three right angle triangles $T_1$ with sides 3, 4, 5, T2 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

Here, $R=\left\{\right(T_1,T_2):T_{1}issimilarto{T}_2\}$
R is reflexive; since every triangle is similar to itself.

Further. if $(T_1,T_2)\in R$, then $T_1$ is similar to $T_2.$
$\Rightarrow T_2$ is similar to $T_1$
Therefore, R is symmetric.
Now, let $(T_1,T_2),(T_2,T_3)\in R.$
$\Rightarrow T_1$ is similar to $T_2$ and $T_2$ is similar to $T_3.$
$\Rightarrow T_1$ is similar to $T_3\Rightarrow (T_1,T_3)\in R$
Therefore, R is transitive. Thus, R is an equivalence relation.
Now, we can observe that $\frac{3}{6}=\frac{4}{8}=\frac{5}{10}=\left(\frac{1}{2}\right)$
Therefore, the corresponding sides of triangles $T_1$ and $T_3$ are in the same ratio. Then, triangle $T_1$ is similar to triangle $T_3.$
hence, $T_1$ is related to $T_3$.