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 $1$ |

| Discuss |

Answer: Option C

Explanation:

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.*

Post Your Comment