Ncert Solutions for Class 10 Maths (Triangles)

Exercise 6.6 (Optional)

1. In Figure, PS is the bisector of ∠QPR of ΔPQR. Prove that \({QS \over SR} = {PQ \over PR} \).

2. In Figure, D is a point on hypotenuse AC of ΔABC, such that BD⊥AC, DM⊥BC and DN⊥AB. Prove that : (i) \(DM^2 = DN.MC\) (ii) \(DN^2 = DM.AN\)

3. In Figure, ABC is a triangle in which ∠ABC > 90° and AD⊥CB produced. Prove that \(AC^2 = AB^2 + BC^2 + 2BC.BD\).

4. In Figure, ABC is a triangle in which ∠ABC < 90° and AD⊥BC. Prove that \(AC^2 = AB^2 + BC^2 – 2BC.BD\).

5. In Figure, AD is a median of a triangle ABC and AM⊥BC. Prove that : (i) \(AC^2 = AD^2 + BC.DM +({BC \over 2})^2 \) (ii) \(AB^2 = AD^2 – BC.DM +({BC \over 2})^2 \) (iii) \(AC^2 + AB^2 = 2 AD^2 +{1 \over 2} BC^2 \).

6. Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

7. In Figure, two chords AB and CD intersect each other at the point P. Prove that : (i) ΔAPC ~ ΔDPB (ii) AP.PB = CP.DP

8. In Figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) ΔPAC ~ ΔPDB (ii) PA.PB = PC.PD

9. In Figure, D is a point on side BC of ΔABC such that \({BD \over CD}={AB \over AC}\). ⋅ Prove that AD is the bisector of ∠BAC.

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 Figure)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?

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