Question

equilateral triangles are drawn on the three sides of a right angled triangle. show that the area of the hypothenuse is equal Ty o the sum of the areas of triangle on the other two sides​

Answer 1

Given A right angled triangle ABC with right angle at B. Equilateral triangles PAB, QBC and RAC are described on sides AB, BC and Ca respectively.

To prove Area(△PAB)+Area(△QBC)=Area(△RAC).

Proof Since triangles PAB, QBC and RAC are equilateral. Therefore, they are equiangular and hence similar.

∴

Area(△RAc)

Area(△PAB)

+

Area(△RAC)

Area(△QBC)

=

AC

2

AB

2

+

AC

2

BC

2

⇒

Area(△RAC)

Area(△PAB)

+

Area(△RAC)

Area(△QBC)

=

AC

2

AB

2

+BC

2

⇒

Area(△RAC)

Area(△PAB)

+

Area(△RAC)

Area(△QBC)

=

AC

2

AC

2

=1

[∵△ is a right angled triangle with ∠B=90

0

∴AC

2

=AB

2

+BC

2

]

⇒

Area(△RAC)

Area(△PAB)+Area(△QBC)

=1

⇒ Area(△PAB)+Area(△QBC)=Area(△RAC) [Hence proved]