Question

In the given figure, L is the midpoint of QR and PL is produced to N. If L is the midpoint of PN and PL bisects angle QPR, then show that: (i)QN || PR (ii)PQ=PR​

Answer 1

Solution :-

Join NR .

In ∆PQR given that,

→ QL = LR { L is the mid point of QR . }

and,

→ ∠QPL = ∠RPL { PL is angle bisector of ∠QPR .}

so,

→ ∆PQR = Isosceles ∆ . { If the bisector of the vertical angle of a ∆ bisects the base , it is an Isosceles ∆. }

then,

→ PL ⟂ QR . { In Isosceles ∆ median is ⟂ to the base . }

therefore,

→ PQ = PR { Since ∆PQR is an Isosceles ∆ . }

now, in quadrilateral PQNR , we can conclude that diagonal PN bisects the diagonal QR at right angle .

As we know that, a quadrilateral whose diagonals bisect each other at right angles is a rhombus and opposite sides of rhombus are parallel to each other .

hence,

→ QN || PR .

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