Question

In triangle PQR if pt is a median then show that PQ+PR>2PT

Answer 1

[FIGURE IS IN THE ATTACHMENT]
Given:
A ∆PQR in which PT is a Median .
To Prove: PR + PQ > 2PT
Construction: produce PT to S , such that PT = TS. Join SR.
PROOF:
In ∆PTQ & ∆STR
PT=ST ( By construction).......(1)
∠PTQ = ∠STR (vertically opposite angles)
QT=RT (PT is a median, T is a midpoint of QR)
∆PTQ≅∆STR (By SAS congruence rule)
PQ= SR (By CPCT)...........(2)
In ∆PSR
PR +SR> PS
[Sum of any two sides of a triangle is greater than the third side]
PR +SR> PT + TS (PS= PT+TS)
PR + PQ >PT + TS (from eq 2)
PR + PQ >PT + PT (from eq 1)
PR + PQ > 2PT
Hence, the sum of any two sides of a triangle is greater than twice the median with respect to the third side.

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