Question

Which of the following lengths and widths represents a rectangle whose diagonal is rational? Question 3 options: length = 2, width = 1 length = 4, width = 4 length = 3, width = 2 length = 4, width = 3

Answer 1

Correct option is length = 4, width = 3.

Explanation:

Given:

Diagonal of a rectangle is rational.

To find:

Which of the following length and width options represent a rectangle ?

options:

length = 2, width = 1

length = 4, width = 4

length = 3, width = 2

length = 4, width = 3

Solution:

First of all, let us consider a rectangle as shown in the attached answer image.

Rectangle ABCD.

Width of rectangle is AB.

Width of rectangle is BC.

And the diagonal AC or BD can be found by using Pythagorean Theorem:

`\text{Hypotenuse}^(2) = \text{Base}^(2) + \text{Perpendicular}^(2)\\\Rightarrow AC^(2) = AB^(2) + BC^(2)\\\Rightarrow Diagonal^(2) = Length^(2) + Width^(2)`

Now, let us find diagonal for each option and check whether it is rational or not.

Option 1:

length = 2, width = 1

`Diagonal^(2) = 2^(2) + 1^(2)\\\Rightarrow Diagonal^(2) = 5\\\Rightarrow Diagonal = \sqrt5`

Not rational

Option 2:

length = 4, width = 4

`Diagonal^(2) = 4^(2) + 4^(2)\\\Rightarrow Diagonal^(2) = 32\\\Rightarrow Diagonal = 4\sqrt2`

Not rational.

Option 3:

length = 3, width = 2

`Diagonal^(2) = 3^(2) + 2^(2)\\\Rightarrow Diagonal^(2) = 13\\\Rightarrow Diagonal = √(13)`

Not rational.

Option 4:

length = 4, width = 3

`Diagonal^(2) = 4^(2) + 3^(2)\\\Rightarrow Diagonal^(2) = 25\\\Rightarrow Diagonal = √(25) = 5`

Diagonal is rational.

Correct option is length = 4, width = 3.