We are asked to write the converse, the inverse, and the contrapositive of the following conditional statement.

" If a figure is a rectangle with sides 2 cm and 3 cm, then it has a perimeter of 10 cm . "

The statement can be written in the following form.

p \to q

Here,

p

is the hypothesis and

q

the conclusion. Let's draw a rectangle with sides

2

cm and

3

cm, to decide whether the statement is True or False.

The perimeter of the above rectangle is $3$ cm $+ 3$ cm $+ 2$ cm $+ 2$ cm $= 10$ cm. Therefore, the statement is True.

Converse

To write the converse of the conditional statement, we need to exchange the hypothesis and the conclusion.

Conditional : Converse : p \to q q \to p

For the given statement, the converse is as follows.

" If a figure has a perimeter of 10 cm, then it is a rectangle with sides 2 cm and 3 cm . "

Let's draw a figure with perimeter

10

cm, to decide whether the statement is True or False.

We can see that the above figure is a rectangle with a perimeter of $10$ cm. But, it does not have sides measuring $2$ cm and $3$ cm. Therefore, we have found a counterexample and this statement is False.

Inverse

To write the inverse of the conditional statement, we need to negate both the hypothesis and the conclusion.

Conditional : Inverse : \neg p \to \neg q \neg p \to \neg q

Hence, for the given statement, the inverse is as follows.

" If a figure is not a rectangle with sides 2 cm and 3 cm, then it does not have a perimeter of 10 cm . "

Let's draw a figure which is not a rectangle with sides

2

cm and

3

cm to decide whether the statement is True or False.

We can see that the above figure is not a rectangle with sides $2$ cm and $3$ cm. However, the perimeter is $3$ cm $+ 3$ cm $+ 4$ cm $= 10$ cm. Therefore, we have found a counterexample and this statement is False.

Contrapositive

To write the contrapositive of the conditional statement, we need to negate both the hypothesis and the conclusion of the converse.

Conditional : Contrapositive : \neg p \to \neg q \neg q \to \neg p

Hence, for the given statement, the contrapositive is as follows.

" If a figure does not have a perimeter of 10 cm, then it is not a rectangle with sides 2 cm and 3 cm . "

We do not need to draw a figure to decide whether this statement is True or False. If a figure does not have a perimeter of

10

cm, it is impossible for it to be a rectangle with sides

2

cm and

3

cm. Hence, this statement is True.

Exercise