Question

Find the area of triangle when ​

Answer 1

Solution :

Let ΔABC be an equilateral triangle and OP, OQ and OR be the perpendiculars from an interior point O. Then join OA, OB and OC.

Let sides of an equilateral triangle be a cm.

Then, according to the question

Area of ΔOAB = ½ x AB x OP

= ½ x a x 14

= 7a cm² ..........(i)

Area of ΔOBC = ½ x BC x OQ

= ½ x a x 10

= 5a cm² .............(ii)

and Area of ΔOAC =½ x AC x OR

= ½ x a x 6

= 3a cm² ............(iii)

Now, area of an equilateral ΔABC

Area of (ΔOAB + ΔOBC +ΔOAC)

= (7a + 5a + 3a) cm²

= 15a cm² ........(iv)

But area of equilateral triangle is √3/4 a²

√3/4 a² = 15 a

a = 15 x 4/√3 x √3/√3 [by rationalising]

= 60√3/3

= 20√3 cm

On putting a = 20√3 in eqn. (iv), we get

Area of an equilateral

ΔABC = 15 x 20 √3

= 300√3 cm²

∴ Hence, the required area of an equilateral ΔABC is 300√3 cm²