In a triangle, PQR, N is a point on PR such that QN PR. If PN. NR = QN2, prove that PQR = 90° .
Given:- In a triangle PQR, N is a point on PR such that QN PR and PN.NR = QN2
To prove:- PQR = 90°
Proof:
We have PN.NR = QN2
and PNQ = RNQ (each equal to 90°)
we know that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the triangles are similar by SAS similarity criterion.
Then QNP and RNQ are equiangular.
i.e.
adding equation (2) and (3) we get
In triangle PQR
[Using equation (4)]
Hence proved