In triangle pqr, pt is a median. Mid point of pt is s. Qs is produced to meet pr at x. If pr=12.75 m find px
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Answer:
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[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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Answer:
Apollonius's theorem relates the length of a median of a triangle to the lengths of its side. It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".
Step-by-step explanation:
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.