write the prove of basic proportionally theorem.p

Question

write the prove of basic proportionally theorem.prove it ........................................................................................

Armaan Arora · Accepted Answer

Basic Proportionality Theorem states that, if a line is parallel to a side of a triangle which intersects the other sides into two distinct points,then the line divides those sides of the triangle in proportion.

Let ABC be the triangle.

The line l parallel to BC intersect AB at D and AC at E.

To prove DBAD=ECAE

Join BE,CD

Draw EF⊥AB, DG⊥CA

Since EF⊥AB,

EF is the height of triangles ADE and DBE

Area of △ADE=21× base × height=21AD×EF

Area of △DBE=21×DB×EF
areaofΔDBEareaofΔADE=21×DB×EF21×AD×EF=DBAD ........(1)
Similarly,
areaofΔDCEareaofΔADE=21×EC×DG21×AE×DG=ECAE ......(2)

But ΔDBE and ΔDCE are the same base DE and between the same parallel straight line BC and DE.

Area of ΔDBE= area of ΔDCE ....(3)

From (1), (2) and (3), we have

DBAD=ECAE

Hence proved.

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