O is any point within a triangle pqr whose sides are 6cm ,7cm and 9cm respectively. Prove that (OP +OQ +OR) > 11cm ?
Give figure also.
Answers
If three sides of triangle PQR is 6 cm, 7 cm & 9 cm and O is any point inside it then (OP+OQ+OR) > 11 cm.
Step-by-step explanation:
Referring to the figure attached below we have,
In ∆PQR,
PQ = 6 cm
QR = 7 cm
PR = 9 cm
Let O be any point inside the triangle and join OP, OQ & OR.
According to the Triangle Inequality Theorem, the sum of any two sides of a triangle must be greater than the measure of the third side.
Therefore,
In ∆POQ, [OP + OQ] > 6 cm ….. (i)
In ∆QOR, [OQ + OR] > 7 cm ….. (ii)
In ∆POR, [OP + OR] > 9 cm ……. (iii)
Now, adding eq. (i), (ii) & (iii), we get
[OP + OQ] + [OQ + OR] + [OP + OR] > [6 + 7 + 9] cm
⇒ 2 [OP + OQ + OR] > [22 cm]
⇒ [OP + OQ + OR] > [] cm
⇒ [OP + OQ + OR] > 11 cm
Hence proved
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