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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 D

Explanation :

A square and a rhombus on the same base will have equal areas.

Hence ratio of the areas of the square and the rhombus will be equal to 1 since
they stand on the same base

Note : Please find the proof of the formula given below which you may like to go through

Let ABCD be the square and ABEF be the rhombus

Consider the right-angled triangles ADF and BCE

We know that AD = BC (∵ sides of a square)

AF = BE (∵ sides of a rhombus)

∵ DF = CE [∵ DF2 = AF2 - AD2 and CE2 = BE2 - BC2]

Hence Δ ADF = Δ BCE

=> Δ ADF + Trapezium ABCF= Δ BCE + Trapezium ABCF

=> Area of square ABCD = Area of rhombus ABEF


Alex 01 May 2014 5:22 AM
I agree, the proof given here is incorrect. 
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P 30 Apr 2014 7:17 AM
AB= AD (sides of a Square)
But AF^2 = DF^2+ AD^2 (Pythagoras theorem)
Or AF = sq. root (DF^2 +AD^2) > AD
Or AF > AB (since AB = AD)
Therefore : AF is not equal to AB 
Which means AFEB is NOT a rhombus. 
The ratio of the areas of a square and rhombus >= 1. 

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