Question

In the given figure, P is the midpoint of YZ and AP = BP. Also AP _| XY and PB _| XZ. Prove that XYZ is an isosceles triangle.​

Answer 1

Construction :-

Join XP

Answer :-

First we need to prove Angle XYZ and Angle XZY as equal for that we need to show Triangle APY and Triangle BPZ as congrent then we can Prove Triangle XYP and Triangle XZP as congrent then by C.P.C.T XY = XZ i.e XYZ is an isosceles triangle. Let's proceed :-

In Triangles APY and Triangles BPZ

AP = BZ [ Given ]

YP = PZ [ P is the midpoint of YZ ]

Angle PAY = Angle PBZ [ Each 90° ]

Triangles congrent by AAS Congrency , so C.P.C.T

Angle XYP = Angle XZP

Now ,

In triangle XYP and Triangle XZP

YP = PZ [ P is the midpoint of YZ ]

XP = XP [ Common ]

Angle XYP = Angle XZP [ C.P.C.T ]

Therefore , Triangles Congrent by AAS Congrency. So ,

XY = XZ [ C.P.C.T ]

As 2 sides equal Therefore XYZ is an isosceles triangle.