Question

State and prove Basic Proportionality theorem
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Answer 1

Basic proportionality theorem: 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. ... Hence we can say that the basic proportionality theorem is proved.

Answer 2

Answer:

Hint: To prove this theorem first we will join BE and CD. Then draw a line EL perpendicular to AB and line DM perpendicular to AC. Now we will find the ratio of area of ΔADE to ΔDBE and ratio of area of ΔADE to ΔECD. Comparing the ratios we will get the final answer.

Complete step-by-step answer:

Now, ΔDBE and ΔECD being on the same base DE and between the same parallels DE and BC, we have,ar(ΔDBE)=ar(ΔECD) then we say that the basic proportionality theorem is proved.

Basic proportionality theorem:

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.

Given:

A ΔABCin which DE∥BCand DE intersects AB and AC at D and E respectively.

To prove that:

AD

DB

=

AE

EC

Construction:

Join BE and CD.

Draw EL⊥ABand DM⊥AC

Proof:

We have the

ar(ΔADE)=

1

2

×AD×EL

ar(ΔDBE)=

1

2

×DB×EL

Therefore the ratio of these two is

ar(ΔADE)

ar(ΔDBE)

=

AD

DB

. . . . . . . . . . . . . . (1)

Similarly,

ar(ΔADE)=ar(ΔADE)=

1

2

×AE×DM

ar(ΔECD)=

1

2

×EC×DM

Therefore the ratio of these two is

ar(ΔADE)

ar(ΔECD)

=

AE

EC

. . . . . . . . . . . .. . . (2)

Now, ΔDBE and ΔECD being on the same base DE and between the same parallels DE and BC, we have,

ar(ΔDBE)=ar(ΔECD). . . . . . . . . . . (3)

From equations 1, 2, 3 we can conclude that

AD

DB

=

AE

EC

Hence we can say that the basic proportionality theorem is proved.

Note: The formula for area of the triangle is given by

1

2

×b×hwhere b, h are base and height respectively. If two triangles are on the same base and between the same parallels then the area of those two triangles are equal.