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Relationship between Unequal Sides of Triangle and the Angles Opposite to It.

S is any poin...

Question

S is any point on side QR of a $Δ P Q R$ . Show that PQ + QR + RP > 2 PS.
(2 Marks)

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Solution

Given in $Δ P Q R$ , S is any point on side QR

$I n Δ P Q S, P Q + Q S > P S$ (0.5 Marks)
[sum of two sides of a trianlge is greater than the third side]
$S i m i l a r l y, i n Δ P R S, S R + P R > P S$
(0.5 Marks)
[sum of two sides of a triangle is greater than the third side]

On adding Eqs (i) and (ii). we get,
$P Q + Q S + S R + R P > 2 P S$
$\Rightarrow P Q + (Q S + S R) + R P > 2 P S \Rightarrow P Q + Q R + R P > 2 P S [∵ Q R = Q S + S R]$ (1 Mark)

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