Question

PQR is a triangle right angled at R. A line through the mid-point M of hypotenuse PQ and

parallel to QR intersects PR at S. Show that

(i) S is the mid-point of PR (ii) MS ⊥ PR (iii) PM = RM​

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

Answer 2

Given,

PQR is a right angled triangle at R.

M hypotenuse to PQ and parallel to QR intersects PR at S.

To show that,

S is the mid point of PR ,MS ⊥ PR and PM = RM

Solution,

In △PQR,

(1) S is the mid point of PQ and

MS∥PQ

S is the mid point of PR

(2) As MS∥PQ &

AR is transversal

∠MSR + ∠QRS = 180°

∠MSR + 90° = 180° ⇒ ∠MSR = 90° ⇒ MS ⊥ PR

(3) In △PMS and △QMS

PS=RS

∠PSM=∠RQM

SM=SM

△PMS≅△RMS

PM=RM

Hence, MS ⊥ PR and PM = RM​ are proved