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 $\dfrac{1}{2}$ B. equal to $\dfrac{3}{4}$ C. greater than $1$ D. equal to $1$

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 square $=a^2\cdots(1)$

From the diagram, $\sin \theta =\dfrac{\text{h}}{a}$
$\Rightarrow h=a\sin\theta$

Area of the rhombus
$=ah=a×a\sin\theta=a^2\sin\theta~~\cdots(2)$

From $(1)$ and $(2)$

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

Since $0° \lt \theta \lt 90°, 0 \lt \sin θ \lt 1$.
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.

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