Question

If the length of the
sides of a square are
rational numbers,
which of the
following is true?
Half of its side
length is always
an integer
Length of its
diagonal is an
irrational
number
Its area is
definitely a
perfect square
number.
Its perimeter
may or may not
be a rational
number.​

Answer 1

b is the correct answer

Answer 2

SOLUTION :

GIVEN

If the length of the sides of a square are rational numbers

TO CHOOSE THE CORRECT OPTION

• Half of its side length is always an integer

• Length of its diagonal is an irrational number

• Its area is definitely a perfect square number.

• Its perimeter may or may not be a rational number.

EVALUATION

Here it is given that the length of the sides of a square are rational number

CHECKING FOR OPTION : 1

Since the length of the sides of a square are rational number

So Half of its side length is always a rational number may not be an integer

For example,

Suppose the length of the sides of a square

$\displaystyle \sf{ = \frac{5}{4} } \: \: unit$

Then Half of its side length

$\displaystyle \sf{ = \frac{5}{8} } \: \: unit$

Which is rational but not an integer

So this option is not correct

CHECKING FOR OPTION : 2

Since the length of the sides of a square are rational number

So Length of its diagonal is always an irrational number

Since Length of its diagonal

$\sf{ = \sqrt{2} \times length \: of \: the \: side}$

This option is CORRECT

CHECKING FOR OPTION : 3

Since the length of the sides of a square are rational number

So area may not be a perfect square number.

For example

Suppose the length of the sides of a square

$\displaystyle \sf{ = \frac{5}{4} } \: \: unit$

Then area

$\displaystyle \sf{ = { \bigg( \frac{5}{4} \bigg)}^{2} } = \frac{25}{16} \: \: \: sq\: \: unit$

Which is not a perfect square ( since it is not an integer)

So this option is not correct

CHECKING FOR OPTION : 4

Since the length of the sides of a square are rational number

So Its perimeter is definitely a rational number.

Since for a square,

perimeter = 4 × length of a side

So this option is not correct

FINAL RESULT

If the length of the sides of a square are rational numbers then

Length of its diagonal is always an irrational number

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