Question

State and prove Basic proportionality theorem​

Answer 1

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=21​AD×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.

Answer 2

Answer:

Basic Proportionality Theorem (BPT) If a side is parallel to one side of a triangle and it intersects the other two points in two distinct points, the it divides the other two sides in proportion.