Qno14:- Sides AB and AC and median AD of a ∆ABC are respectively proportional to sides PQ and PR and median PM of another ∆PQR۔ Show that ∆ABC ~PQR۔
mathslab98.blogspot.com Qno14:- Sides AB and AC and median AD of a ∆ABC are respectively proportional to sides PQ and PR and median PM of another ∆PQR۔ Show that ∆ABC ~PQR۔ Given: AB/PQ=AC/PR=AD/PM۔ and In ∆ABC, AD is the median. • BD = DC Also in ∆PQR, PM is the median. • QM= MR۔ Construction: in ∆ABC Draw DG || AC and DE || AB۔ Also in ∆PQR Draw FM || PR and HM|| PQ۔ To Prove:- ∆ABC ~ ∆PQR۔ Proof:- Since D is the mid point of BC and DG || AC. So by Converse of Mid Point Theorem. The converse of midpoint theorem states that: "If a line segment is drawn passing through the midpoint of any one side of a triangle and parallel to another side, then this line segment bisects the remaining third side. i,e DG bisects AB. => AG = BG Similarly AE = EC Also in ∆PQR ; PF = FQ and PH = HR Now in ∆ABC ; GD || AE and AG|| DE => Quadrilateral AGDE is a Parallelogram . Then GE = AE ..........1. ( Opposite sides of ||gm are equal) similarly in ∆PQR ; F
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