Question

Let $T$ be the set of all triangles in a plane and a relation $R$ on $T$ be defined by $xRy\iff x$ is similar to $y$ i.e., $R=\{(x$, $y)$ ; $x$ is similar to $y\}$. Show that $R$ is an equivalence relation on $T$. Consider three right angled triangles : $x$ with sides $3$, $4$, $5$; $y$ with sides $5$, $12$, $13$ and $z$ with sides $6$, $8$, $10$, which triangles among $x$, $y$ and $z$ are related?

• A. $x$ and $y$
• B. $y$ and $z$
• C. $x$ and $z$
• D. Both $(a)$ and $(b)$

Answer 1

Answer: (C)

$(i)$ Every triangle is similar to itself.
$\therefore x$ is similar to $x$, $\forall x\in T$. i.e., $xRx$, $\forall x\in T$
So $R$ is reflexive on $T$.
$(ii)$ Let $xRy\Rightarrow x$ is similar to $y\Rightarrow y$ is similar to $x$.
$\Rightarrow yRx\therefore$ $R$ is symmetric on $T$.
$(iii)$ Let $xRy$ and $yRz\Rightarrow x$ is similar to $y$ and $y$ is similar to $z$
$\Rightarrow x$ is similar to $z\Rightarrow xRz$.
$\therefore R$ is transitive on $T$.
Thus $R$ is an equivalence relation on $T$.
Now, since sides of triangles $x$ and $y$ are not proportional therefore $x/Ry$. But the sides of triangles $x$ and $z$ are proportional, therefore $xRz$ . Also $y/Rz$.