Question

state and proove basic proportionality theoram ​

Answer 1

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

DB

AD

=

EC

AE

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Join BE,CD

Draw EF⊥AB, DG⊥CA

Since EF⊥AB,

EF is the height of triangles ADE and DBE

Area of △ADE=

2

1

× base × height=

2

1

AD×EF

Area of △DBE=

2

1

×DB×EF

areaofΔDBE

areaofΔADE

=

2

1

×DB×EF

2

1

×AD×EF

=

DB

AD

........(1)

Similarly,

areaofΔDCE

areaofΔADE

=

2

1

×EC×DG

2

1

×AE×DG

=

EC

AE

......(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

DB

AD

=

EC

AE

Hence proved.