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IN A TRIANGLE ABC IT IS GIVEN THAT XY IS PARALLEL TO AC AND IT DIVIDES THE TRIANGLE INTO TWO PARTS OF EQUAL AREA . FIND AX DIVIDED BY AB

If a line is drawn parallel to a side of a triangle, then the sides of the new triangle formed are proportional to the sides of the given triangle                                                                               

(New triangle is ‘similar’ to the given triangle)

Data: XY is parallel to BC

To prove: AX/AB=AY/AC=XY/BC

Construction: Draw a line parallel to AC from X and let that line meet BC at Z.

Step

Statement

Reason

1

BZ/BC=BX/BA

BPT since XZ || AC (Construction)

2

XY=ZC, YC =XZ

By construction, XZCY is a parallelogram and hence opposite sides are equal

3

AX/AB =AY/AC 

BPT since XY || BC (Given)

4

LHS =AX/AB = (AB-BX)/AB

X is a point on AB

5

= 1-(BX/AB)

 

6

= 1-(BZ/BC)

From (1)

7

= (BC-BZ)/BC

Z is a point on BC

8

=ZC /BC

 

9

=XY/BC=AY/AC

From(2)


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