Option d: a rhombus with a side length [tex]\sqrt{5}[/tex] units and diagonals with lengths [tex]\sqrt{2}[/tex] units and [tex]\sqrt{18}[/tex] units
Explanation:
The formula to find the area of the rhombus is [tex]A=\frac{p q}{2}[/tex] where p,q are diagonals.
The formula to find the perimeter of the rhombus is [tex]P=4 a[/tex] where a is the side length.
Now, we shall determine the rhombus which have a rational area and an irrational perimeter.
Option a: a rhombus with a side length 5 units and diagonals with lengths 8 units and 6 units
Area = [tex]\frac{8 \times 6}{2}=\frac{48}{2}=24[/tex]
Perimeter = [tex]4(5)=20[/tex]
This is a rhombus with a rational area and a rational perimeter.
Hence, Option a is not the correct answer.
Option b: a rhombus with a side length [tex]\sqrt{3}[/tex] units and diagonals with lengths [tex]\sqrt{3}[/tex] units and [tex]\sqrt{9}[/tex] units
Area = [tex]\frac{\sqrt{3} \times \sqrt{9}}{2}=2.59807 \ldots \ldots[/tex]
Perimeter = [tex]4(\sqrt{3} )=6.92820......[/tex]
This is a rhombus with irrational area and irrational perimeter.
Hence, Option b is not the correct answer.
Option c: a rhombus with a side length [tex]\sqrt{5}[/tex] units and diagonals with lengths [tex]\sqrt{2}[/tex] units and [tex]\sqrt{18}[/tex] units
Area = [tex]\frac{\sqrt{2} \times \sqrt{18}}{2}=\frac{\sqrt{36}}{2}=3[/tex]
Perimeter = [tex]4(\sqrt{5} )=8.94427......[/tex]
This is a rhombus with rational area and irrational perimeter.
Hence, Option c is the correct answer.
Option d: a rhombus with a side length 2.5 units and diagonals with lengths [tex]\sqrt{5}[/tex] units and [tex]\sqrt{20}[/tex] units
Area = [tex]\frac{\sqrt{5} \times \sqrt{25}}{2}=5.59016 \ldots \ldots[/tex]
Perimeter = [tex]4(2.5)=10[/tex]
This is a rhombus with an irrational area and a rational perimeter.
Hence, Option d is not the correct answer.
