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4Prove that if a line bisects any interior angle of a triangle, then this line divides the lineopposite to that angle in the ratio of length of remaining sides of triangle. |
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Answer» Let ABC be the triangle AD be INTERNAL BISECTOR of ∠BAC which meet BC at D To prove: DC BD
= AC AB
DRAW CE∥DA to meet BA produced at E Since CE∥DA and AC is the transversal. ∠DAC=∠ACE (alternate angle ) .... (1) ∠BAD=∠AEC (corresponding angle) .... (2) Since AD is the angle bisector of ∠A ∴∠BAD=∠DAC .... (3) From (1), (2) and (3), we have ∠ACE=∠AEC In △ACE, ⇒AE=AC (∴ Sides opposite to equal angles are equal) In △BCE, ⇒CE∥DA ⇒ DC BD
= AE BA
....(Thales Theorem) ⇒ DC BD
= AC AB
....(∴AE=AC) |
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