Question

State and prove basic proportionality theorem.

Thanks....​

Answer 1

→ Theorem

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

→

Given :- ΔABC, BC ∥ DE, DE intersects sides AB and AC.

To prove that :- $\frac{AD}{DB}=\frac{AE}{EC}$

Construction :- Join BE and CD, draw DM ⊥ AC

Proof :- We know that are of triangle = 1/2 x b x h

- Area of ΔADE = 1/2 x AD x EN

similarly,

- Area of ΔBDE = 1/2 x DB x EN

and,

- Area of ΔDEC = 1/2 x EC x DM

-----------------

Therefore,

$\frac{Ar(ADE)}{Ar(BDE)} =\frac{1/2*AD*EN}{1/2*DB*EN} =\frac{AD}{DB} -(1)$

Similarly,

$\frac{Ar(ADE)}{Ar(DEC)} =\frac{1/2*AE*DM}{1/2*EC*DM} =\frac{AE}{EC} -(2)$

---------------

ΔBDE and ΔDEC are on the same bases and between the same parallel DE, Therefore their area will be equal.

- arΔBDE = arΔDEC

--------------

From (1) and (2)

$\frac{AD}{DB}=\frac{AE}{EC}$

↬ Proved.

Answer 2

$\huge\underline\mathfrak\red{Statement}$

★ If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio or are proportional. ★

____________________

$\huge\underline\mathfrak\red{Solution}$

Given : In ∆ABC, PQ || BC

To prove : AP / PB = AQ / QC

Construction : Draw QH perpendicular to AB, Join BQ and PC. [ You can take one more perpendicular ].

Proof : Area of ∆APQ = 1/2 × base × height

= 1/2 × AP × QH

Similarly, Area ∆PBQ = 1/2 × base × height

= 1/2 × PB × QH

___________________

Ratio of their areas :

Area ∆APQ / Area ∆PBQ = AP / PB ( 1/2 , 1/2 and QH, QH are cancelled out ) _____(1)

Similarly,

Area ∆APQ / Area ∆PQC = AQ / QC _____(2)

Now,

∆PBQ and ∆PQC are on the same base and lying between same parallel lines,

Therefore,

Area ∆PBQ = Area ∆PQC _____(3)

___________________

From (1), (2) and (3) :

We can write : AP / PB = AQ / QC

___________________

Therefore,

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points,then the other two sides are divided in the same ratio or are proportional.

___________________

Hence proved!