Question

of their corresponding medians.
7. Prove that the area of an equilateral triangle described on one side of a square is equ
to half the area of the equilateral triangle described on one of its diagonals.​

Answer 1

hi friend,

here is your answer

Answer:

the area of an equilateral triangle described on one side of a square is equ

to half the area of the equilateral triangle described on one of its diagonals.

Step-by-step explanation:

Given:

ABCD is a Square,

DB is a diagonal of square,

△DEB and △CBF are Equilateral Triangles.

To Prove:

A(△DEB)

A(△CBF)

=

2

1

Proof:

Since, △DEB and △CBF are Equilateral Triangles.

∴ Their corresponding sides are in equal ratios.

In a Square ABCD, DB=BC

2

.....(1)

∴

A(△DEB)

A(△CBF)

=

4

3

×(DB)

2

4

3

×(BC)

2

∴

A(△DEB)

A(△CBF)

=

4

3

×(BC

2

)

2

4

3

2×(BC)

(From 1)

∴

A(△DEB): A(△CBF)

=2: 1

hence, proved

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Answer 2

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