12. If a square and a rhombus stand on the same base, then what is the ratio of the areas of the square and the rhombus? A. equal to ½ B. equal to ¾ C. greater than 1 D. equal to 1

Here is the answer and explanation

Answer : Option C

Explanation :

If a square and a rhombus lie on the same base, area of the square will be greater than area of the rhombus (In the special case when each angle of the rhombus is 90°, rhombus is also a square and therefore areas will be equal)

Hence greater than 1 is the more suitable choice from the given list

================================================================
Note : Proof

Consider a square and rhombus standing on the same base 'a'. All the sides of a square are of equal length. Similarly all the sides of a rhombus are also of equal length. Since both the square and rhombus stands on the same base 'a',

Length of each side of the square = a
Length of each side of the rhombus = a

Area of the sqaure = a2 ...(1)

From the diagram, sin θ = $\dfrac{\text{h}}{a}$
=> h = a sin θ

Area of the rhombus = ah = a × a sin θ = a2 sin θ ...(2)

From (1) and (2)

$\dfrac{\text{Area of the square}}{\text{Area of the rhombus}}=\dfrac{\text{a}^2}{\text{a}^2 \sin θ} = \dfrac{1}{\sin \text{θ}}$

Since 0° < θ < 90°, 0 < sin θ < 1. Therefore, area of the square is greater than that of rhombus, provided both stands on same base.

(Note that, when each angle of the rhombus is 90°, rhombus is also a square (can be considered as special case) and in that case, areas will be equal.