Let \(\displaystyle D\) be the length of the shorter diagonal. If the longer diagonal is 20% longer, then it measures 120% of the length of the shorter diagonal; this is 

\(\displaystyle \frac{120}{100} = \frac{120 \div 20}{100 \div 20} = \frac{6}{5}\)

of \(\displaystyle D\), or \(\displaystyle \frac{6}{5}D\).

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up an equation and solve for \(\displaystyle D\):

\(\displaystyle \frac{1}{2} \cdot \frac{6}{5}D \cdot D = N\)

\(\displaystyle \frac{3}{5}D^{2} = N\)

\(\displaystyle \frac{5}{3} \cdot \frac{3}{5}D^{2} = \frac{5}{3} \cdot N\)

\(\displaystyle D^{2} = \frac{5N}{3}\)

\(\displaystyle D =\sqrt{ \frac{5N}{3}}\)

A quadrilateral is any two-dimensional shape with  \(\displaystyle 4\) sides. The only shape listed that does not have \(\displaystyle 4\) sides is a triangle. 

The only difference between a rectangle and a square is their side lengths. A square has to have \(\displaystyle 4\) equal side lengths, but the opposite side lengths of a rectangle only have to be equal. 

Out of the choices given, the only characteristic used to describe shapes is the number of sides. A triangle has \(\displaystyle 3\) sides and a rectangle has \(\displaystyle 4\) sides. 

By definition, the only two quadrilaterals that have to have \(\displaystyle 4\) right angles, are the square and the rectangle. 

A quadrilateral is a \(\displaystyle 4\) sided shape. The only shape listed that does not have \(\displaystyle 4\) sides is a hexagon, which has \(\displaystyle 6\) sides. 

A rectangle, square, and rhombus can all be classified as a parallelogram because each shape has opposite side lengths that are equal. A kite does not. 

By definition, an isosceles trapezoid has to have \(\displaystyle 2\) equal base angles, but a trapezoid does not have to have equal angles. 

A quadrilateral has to have \(\displaystyle 4\) sides, a circle does not have any sides. 

A parallelogram can not be classified as a square because a square has to have \(\displaystyle 4\) equal sides, but a parallelogram can have two different side lengths, as long as the opposite side lengths are equal. 

A parallelogram can not be classified as a rectangle because a rectangle has to have \(\displaystyle 90^{\circ}\) angles, and a parallelogram does not. 

A parallelogram can not be classified as a triangle because a parallelogram has to have \(\displaystyle 4\) sides and a triangle only has \(\displaystyle 3\) sides. 

A parallelogram can not be classified as a rhombus because a rhombus has to have \(\displaystyle 4\) equal sides and a parallelogram does not. 

A parallelogram has four sides, and all shapes with four sides are quadrilaterals.