Equilateral triangles are drawn on the three sides of a right angled triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.

Question

Philoid · Accepted Answer

Given, right angled triangle ABC with AC as hypotenuse

⇒ Let AB = a, BC = b, AC = c

⇒ we have a² + b² = c² …………(1)

⇒ We know that area of equilateral triangle =

⇒ Area of ACD =

⇒ Area of BCF =

⇒ Area of AEB =

⇒ Area of AEB + Area of BCF =

⇒ From eq(1) we have a² + b² = c²

⇒ Area of AEB + Area of BCF = = Area of ACD

Hence, the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.