Question

Show that the relation $R$ defined in the set $A$ of all triangles as $R=\left\{\right(T_1,T_2):T_1\text{is similar to}{T}_2\}$, 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

$R=\left\{\right(T_1,T_2):T_1\text{is similar to}{T}_2\}$
$R$ 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 $T_1$.
$\Rightarrow (T_2,T_1)\in R$
$\therefore R$ 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:
$\frac{3}{6}=\frac{4}{8}=\frac{5}{10}(=\frac{1}{2})$
$\therefore$ 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 $T_3$.