D is a point on side B C of ∆ A B C such that B D C D = A B A C . Prove that A D is the bisector of ∠ B A C .

Construct a line C E parallel to D A which meets B A produced at E . Therefore, ∠ B A D = ∠ B E C (Corresponding angles)......(1) ∠ D A C = ∠ A C E (Alternate angles)......(2) In Δ D B A and ΔCBE , B D C D = A B A C (Given) ......(3) B D C D = B A A E (Basic proportionality theorem) ......(4) From (3) and (4), A E = A C Therefore, ∠ A C E = ∠ B E C ......(5) So, from (1), (2) and (5) ⇒ ∠ B A D = ∠ D A C Therefore, A D is angle bisector of ∠ B A C .

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MATHEMATICS Textbook for Class X>Chapter 6 - Triangles>Exercise 6.6>Q 8

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In the figure, two chords $A B$ and $C D$ of a circle intersect each other at the point $P$ (when produced) outside the circle. Prove that
$P A . P B = P C . P D$

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MATHEMATICS Textbook for Class X>Chapter 6 - Triangles>Exercise 6.6>Q 9

$D$ is a point on side $B C$ of $∆ A B C$ such that $\frac{B D}{C D} = \frac{A B}{A C}$ . Prove that $A D$ is the bisector of $∠ B A C$ .

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MATHEMATICS Textbook for Class X>Chapter 6 - Triangles>Exercise 6.6>Q 10

Nazima is fly fishing in a stream. The tip of her fishing rod is

1.8 m

above the surface of the water and the fly at the end of the string rests on the water

3.6 m

away and

2.4 m

from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have out (see the given figure)? If she pulls in the string at the rate of