Question

In a parallelogram pqrs, PQ is equals to 12 cm and PS is equals to 9 cm. The bisectors of angle P meets SR in M. PM and QR both when produce meet at T. Find the length of RT.

Answer 1

Parallelogram PQRS is given with PQ=12 cm and PS=9 cm.

PM is the angle bisector of angle P, that means

Angle MPQ = Angle MPS = x (let).

Angle MPQ = Angle TMR = x [corresponding angles].

Angle TMR = Angle PMS =x [Vertically opposite angles].

i.e. Angle SPM = Angle PMS = x.

This proves that Triangle SPM is isosceles triangle.

hence SP = SM = 9 cm and MR =3 cm.

Angle SPM = Angle MTR = x [alternate interial angle].

We got Angle TMR = Angle MTR = x. Therefore triangle MTR is also isosceles triangle which means MR = TR = 3 cm.

Hence TR = 3 cm.

Hope this helps!!!!!❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤

Answer 2

$\textbf{\huge{ANSWER:}}$

Given:

PQ = 12 cm
PS = 9 cm
PM is the bisector of Angle P

You may see the attached pic for better understanding.

Now,

As PQ||SR ( PQRS is a ||gm ),
Angle QPM = Angle PMS ( Alt. interior angles )

As Angle MPS = QPM = PMS ( Given and proved above ):
Triangle PSM is isosceles, as 2 angles of the triangle are equal.
Thus, PS = SM = 9 cm
MR = 12 - 9 = 3 cm

Now,

As PS||QT,
Angle MPS = Angle QTP ( Alt. interior angles )

As Angle QPT = MPS = QTP ( Given and proved above ):
Triangle QPT is isosceles, as 2 angles of the triangle are equal.
Thus, PQ = QT = 12 cm
RT = 12 - 9 = 3 cm

Therefore, RT = 3 cm

Hope it Helps!! :)