Question

A rectangle has side lengths of a and b, which are non-zero integers. Determine possible values of a and b, which will make the diagonal of the rectangle a rational number.

A) a=1, b=1
B) a=4, b=5
C) a=3, b=4
D) a=1, b=2

Answer 1

Look at the picture.

Use the Pythagorean theorem:

`a^2+b^2=d^2`

A) a = 1, b = 1

`d^2=1^2+1^2\\\\d^2=1+1\\\\d^2=2\to d=\sqrt2\ \text{it's not rational number}`

B) a = 4, b = 5

`d^2=4^2+5^2\\\\d^2=16+25\\\\d^2=41\to d=√(41)\ \text{it's not rational number}`

C) a = 3, b = 4

`d^2=3^2+4^2\\\\d^2=9+16\\\\d^2=25\to d=√(25)\to d=5\ \boxed{:)}`

D) a = 1, b = 2

`d^2=1^2+2^2\\\\d^2=1+4\\\\d^2=5\to d=\sqrt5\ \text{it's not rational number}}`

Answer: C) a = 3, b = 4.