Question

O is any point in the interior of an equilateral triangle ABC.OP,OQ and OR are perpendicular to sides BC,CA and AB respectively. Find area of triangle ABC

Answer 1

Step-by-step explanation:

Let ABC be the triangle and O a point within ABC. Join OP and produce it to meet BC in D.

In triangle ABD, O lies on AD. Since BP lies between BD and BA, BD<OB<AB …(1)

In triangle ACD, O lies on AD. Since CP lies between CD and CA, CD<OC<AC …(2)

Add (1) and (2) to get

BD<OB<AB

CD<OC<AC

BD+CD < OB+OC < AB+AC. Therefore

AB+AC > OB+OC. Proved.

Answer 2

Let ABC be the triangle and O a point within ABC. Join OP and produce it to meet BC in D.

In triangle ABD, O lies on AD. Since BP lies between BD and BA, BD<OB<AB ... (1)

In triangle ACD, O lies on AD. Since CP lies between CD and CA, CD<OC<AC (2)

Add (1) and (2) to get

BD<OB<AB

CD<OC<AC

BD+CD < OB+OC <AB+AC. Therefore

AB+AC > OB+OC. Proved.