Q8 of 28 Page 107

In the given figure bisector PS of ∠P intersects side QR at S. SN ┴ PQ and SM ┴PR have been drawn. Prove that SN = SM.


Given that PS is the bisector of ∠P intersects side QR at S.

Also, SN ┴ PQ and SM ┴PR


So, ∠NPS=∠MPS…………(1)


And ∠PNS=∠PMS ……………..(2)


In ∆PNS and ∆PMS


∠NPS+∠PNS+∠PSN=180°


And ∠MPS+∠PMS+∠PSM=180°


From (1) and (2)


180°-∠PSN=180°-∠PSM


⇒ ∠PSN=∠PSM………….(3)


Now, in ∆PNS and ∆PMS,


PS=PS (common)


∠NPS=∠MPS (given)


∠PSN=∠PSM (from (3))


∴ ∆PNS ∆PMS (by ASA rule)


So, NS=MS (by cpct)


Hence, proved.


More from this chapter

All 28 →
6

In the given figure on the common base BC there are two isosceles triangles ΔPBC and ΔQBC on the opposite sides of BC. Prove that the line joining the points P and Q bisects line BC at right angles.


 7

In the given figure two isosceles triangle PQR and SQR lie on the same side of the common base QR. Prove that line SP is the perpendicular bisector of line QR.


 9

In the given figure ∠ABC is given. Find the locus of points equidistant from BA and BC and lying in the interior part of ∠ABC.


 1

Find the locus of point ℓ equidistant from three vertices and three sides of a triangle.