Question

PQR is a right-angled triangle at R. A line through the midpoint M of hypotenuse PQ and parallels to QR intersects PR at M. Show that N is the midpoint of PR. MN is perpendicular to PR. RM=MP=1/2PQ

Answer 1

Step-by-step explanation:

In triangles PQS and PQR

∠P is common to both.

and ∠PSQ=∠PQR

Triangles PQS and PQR are equiangular

∴PQPS=QRQS

or, PS=QRPQ⋅QS ...(i)

Again, triangles QRS and PQR are equiangular

∴QRSR=PQQS

or, SR=PQQS⋅QR ...(ii)

From eqns. (i) and (ii)

SRPS=QRPQ⋅QS⋅QR⋅QSPQ=QR2PQ2