Question

o is a point in the interior of triangle pqr , prove that op+oq+or is greater than (pq+1r+pr)

answer and enjoy 20 points

Answer 1

PQ < OP + OQ ( Sum of 2 sides of triangle is greater than 3)
PR < OP + OR ( Sum of 2 sides of triangle is greater than 3)
QR < OP + OR ( Sum of 2 sides of triangle is greater than 3)

NOW ON ADDING LHS AND RHS WE GET -

PQ+ AC + BC < OP + OQ + OP + OR
OQ+ ORPQ + PR + QR< 2OP + 2OQ + 2OR
PQ + PR + QR< 2(OP + OQ + OR)
FINAL ANS = 1/2(PQ + PR + QR) < OP + OQ + OR

HENCE PROVED

Answer 2

PQ < OP + OQ ( Sum of 2 sides of triangle is greater than 3)
PR < OP + OR ( Sum of 2 sides of triangle is greater than 3)
QR < OP + OR ( Sum of 2 sides of triangle is greater than 3)

NOW ON ADDING LHS AND RHS WE GET -

PQ+ AC + BC < OP + OQ + OP + OR
OQ+ ORPQ + PR + QR< 2OP + 2OQ + 2OR
PQ + PR + QR< 2(OP + OQ + OR)
FINAL ANS = 1/2(PQ + PR + QR) < OP + OQ + OR

HENCE PROVED