Question

PROVE THAT THE AREA OF AN EQUILATERAL TRIANGLE DESCRIBED ON ONE SIDE OF THE SQUARE IS EQUAL TO HALF THE AREA OF THE EQUILATERAL TRIANGLE ON ONE OF ITS DIAGONAL​

Answer 1

Step-by-step explanation:

Solution:  

Areas of Similar Triangles NCERT Solutions Tenth Grade

Given: ABCD is a square, AEB is an equilateral triangle described on the side of the square, DBF is an equilateral triangle described on diagonal BD of square.

 

To Prove:     ar(△DBF)ar(△AEB)=21

 

Proof:  Any two equilateral triangles are similar because all angles are of 60 degrees.

Therefore, by AAA similarity criterion, △DBF ~ △AEB

 

ar(△DBF)ar(△AEB)=DB2AB2        (1)

 

{The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides}

 

But, we have DB=2–√AB     {Diagonal of square is 2–√ times of its side}         (2)

 

Putting equation (2) in equation (1), we get

ar(△DBF)ar(△AEB)=(2√AB)2AB2=2AB2AB2 = 2

 

Therefore, area of equilateral triangle described on one side os square is equal to half the area of the equilateral triangle described on one of its diagonals.

Hence Proved