Question

Pls give me answer of 2 maths class 10 chapter 6
( triangles)

Answer 1

We need to do this by similarity :)

To prove PO : PT = PN : PQ

In Δ PQT and ΔPON ,

∠NPO = ∠QPT [ Common ] ...............................(1)

Given :

SO = TS [ given ]

SQ = SR [ given ]

Then ROQT must be a parallelogram because the diagonals of the quadrilateral are bisected .

OR ║ QT [ Opposites sides of parallelogram are parallel ]

This means that ON ║ QT  

[ OR ║ QT and ON is a continuation of OR ]

So , ∠PQT = ∠PNO [ corresponding angles as ON ║ QT ] ....................(2)

This means that :

Δ PQT ≈ Δ PNO [ A.A criteria ( see 1 and 2 ) ]

PO : PT = PN : PQ ...............................(3)

[ corresponding side's ratio of similar triangles ]

To prove NM ║ QR

In Δ PRT and ΔPOM ,

∠MPO = ∠RPT [ Common ] ...............................(4)

ROQT is a parallelogram

TR ║ QO [ Opposites sides of parallelogram are parallel ]

This means that TR ║ OM

[ TR ║ QO and OM is a continuation of QO]

So , ∠PRT = ∠PMO [ corresponding angles as TR ║ OM ] ....................(5)

This means that :

Δ PRT ≈ Δ PMO [ A.A criteria ( see 4 and 5 ) ]

Hence PO : PT = PM : PR ......................(6)

[ corresponding sides of similar triangles ]

From (3) and (6)

PO : PT = PN : PQ ==============> (3)

PO : PT = PM : PR ==============> (6)

Hence : PN : PQ = PM : PR

or , PN/PQ = PM/PR ..........................(7)

In Δ PNM and Δ PQR ,

∠NPM = ∠QPR [ Common ]

PN/PQ = PM/PR [ See (7) ]

So, Δ PNM ≈ ΔPQR [ S.A.S criteria ]

Hence ∠PNM = ∠PQR [ all ∠s of a similar triangle are equal ]

∴ NM║QR [ corresponding ∠s are equal ]

Hope it helps u :)

If u have doubts ask me in the comments

_____________________________________________________________________