Question

Sides of a square are rational numbers, which of the following is true?

a) Half of its side length is always an integer.
b) Length of its diagonal is an irrational number.
c) Its area is definitely a perfect square number.
d) Its perimeter may or may not be a rational number.

Answer 1

Assuming a square has a rational side length 'a', it's not necessary that half of its side length is always an integer, its diagonal's length is always irrational, or its area is always a perfect square. However, it's correct that its perimeter may or may not be a rational number, which aligns with 'a' being rational.

Step-by-step explanation:

The properties of a square can be analyzed to determine the validity of the given options. Let's consider a square with side length 'a', which is a rational number:

1. Half of its side length is always an integer: This is not necessarily true. For example, if 'a' is 3/2, then half of 'a' is 3/4 which is not an integer.
2. Length of its diagonal is an irrational number: This is not always true. For example, if 'a' is 1, the diagonal length is √2, an irrational number. But if 'a' = √2, the diagonal length is 2, a rational number.
3. Its area is definitely a perfect square number: This is not always true. The area of a square is a^2, so it isn't always a perfect square. For instance, if 'a' = 2, then the area is 4, a perfect square, while if 'a' = √2, the area is 2, not a perfect square.
4. Its perimeter may or may not be a rational number: This is true. The perimeter of a square is 4a, it will be a rational number assuming 'a' is a rational number.

Learn more about Properties of a Square