Question

Prove that the equilateral triangles described on the two sides of a right angled triangle
are together equal to the equilateral triangle on the hyplotenuse in terms of their
area ​

Answer 1

Step-by-step explanation:

ANSWER

Given:

△ABC in which ∠ABC=90

∘

and AB=BC. △ABD and △ACE are equilateral triangles.

To Prove: ar(ΔABD)=

2

1

×ar(ΔCAE)

Proof:

Let AB=BC=x units.

∴ CA=

x

2

+x

2

=x

2

units.

Now, ΔABD and ΔCAE being equilateral, has each angle equal to 60

∘

∴ΔABD∼ΔCAE

But, the ratio of the areas of two similar triangles is equal to the ratio of the

squares of their corresponding sides.

∴

ar(ΔCAE)

ar(ΔABD)

=

CA

2

AB

2

=

(x

2

)

2

x

2

=

2x

2

x

2

=

2

1

Hence, ar(ΔABD)=

2

1

×ar(ΔCAE)