KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6

Students can Download Class 10 Maths Chapter 2 Triangles Ex 2.6 Questions and Answers, Notes Pdf, KSEEB Solutions for Class 10 Maths helps you to revise the complete Karnataka State Board Syllabus and score more marks in your in examinations

Karnataka State Syllabus Class 10 Maths Chapter 2 Triangles Ex 2.6

Question 1.
In Fig. 2.56, PS is the bisector of ∠QPR of ∆ PQR, Prove that \(\frac{\mathbf{Q S}}{\mathbf{S R}}=\frac{\mathbf{P Q}}{\mathbf{P R}}\)
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 1
Answer:
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 2
Given: In the figure, PS is the bisector of

KSEEBKSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 3 Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 3

KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6

Question 2.
In Fig. 2.57, D is a point on hypotenuse AD of ∆ ABC, such that BD ⊥ AC, DM ⊥ BC and DN ⊥ AB. Prove that:
i) DM2 = DN.MC ii) DN2 = DM.AN
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 4

Answer:
Given: D is a point on hypotenuse AC of ∆ ABC DM ⊥ BC, DN ⊥AB and BD ⊥ AC
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 5

ii) Consider ∆le ADN And DBN
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 6

Question 3.
In Fig. 2.58, ABC is a triangle in which ∠ABC > 90° and AD ⊥ CB produced. Prove that AC2 = AB2 + BC2 + 2BC. BD
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 7
Answer:
Given: In figure, ABC is a triangle in which ∠ ABC > 90° and AD ⊥ BC.
To prove: AC2 = AB2 + BC2 + 2BC. BD.
Proof: In right triangle ABC
∠D = 90°
AC2 = AD2 + CD2 [Pythagoras theorem]
AC2 = AD2 + (BD + CB)2 [DC = BD + BC]
AC2 = AD2 + BD2+ BC2 +2.BD. BC
AC2 = AB2 + BC2 + 2BD.BC [In right angle ∆le ADB AB2 = AD2 + BD2]

Question 4.
In Fig. 2.59, ABC is a triangle in which ∠ ABC < 90° and AD ⊥ BC. Prove that AC2 = AB2 + BC2 – 2BC. BD.
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 8
Answer:
Given: In figure, ABC is a triangle in which ∠ABC < 90° and AD ⊥ BC
To prove: AC2 = AB2 + BC2 – 2BC.BD
Proof: In right triangle ABC ∠ D = 90°
AC2 = AD2 + DC2 [By pythagoras theorem]
= AD2 + (BC – BD)2 [CD = BC – BD]
= AD2 ± BC2 + BD2 – 2BC. BD
AC2 = AB2 + BC2 – 2BC.BD
[In right angle ADB with
∠D = 90° AB2 = AD2 + BD2 By Pythagoras theorem]

KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6

Question 5.
In Fig. 2.60,AD is a median of a triangle ABC and AM I BC. Prove that:
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 9
Answer:
Given: In figure, AD is a median of a triangle ABC and AM ⊥ BC.
∴ D is mid-point of BC (∵ AD is median)
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 10
Proof:
i) In right triangle AMC ∠M = 90°
∴ AC2 = AM2 + MC2 [By Pythagoras theorem]
= AM2 +(MD + DC)2
= AM2 + MD2 + DC2 + 2MD. DC
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 11
[In right angle ∆le AMD with ∠M = 90°
AM2 + MD2 = AD2 (By pythagoras theorem)]
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 12
ii) In right triangle AMB ∠M = 90°
AB2 = AM2 + MB2 [Pythagoras
theorem]
= AM2 + (BD – MD)2
= AM2 + BD2 + MD2 – 2BD . MD
= AM2 + MD2 + BD2 – (2 BD)MD
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 13

Question 6.
Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
Answer:
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 14

Given: In parallelogram ABCD
AB = CD & AD = BC
Construction: convert parallelogram into a rectangle and Draw AG ⊥ CD
To prove:
AC2 + BD2 = AB2 + BC22 + CD2 – AD2
Proof: consider ∆le BDF ∠D = 90°
BD2 = BF2 + FD2 = h2 + (x+d)2 — (1)
Consider ∆le AGC ∠G = 90°
AC2 = AG22 + GC2 = h2 + (x – d)2 — (2)
Adding (I) and (2)
AC2+ BD2 = h2 + (x + d)2 + h2 + (x – d)2
2h2 + x2 + 2xd + d2 + x2 – 2xd + d2
2h2 + 2x2 + 2d2
= 2x2 + 2(h2 + d2)
= 2x2 + 2y2
= x2 + y2 + x2 + AC2 + BD2 =AB2 +BC2 i-CD2 + AD2.

Question 7.
In Fig. 2.61, two chords AB and CD intersect each other at the point P.
Prove that:
i) ∆ APC ~ ∆ DPB
ii) AP. PB = CP.DP
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 15
Answer:
Given: In figure, two chords AB and CD intersect each other at the point P.
To prove: i) ∆ APC ~ ∆ DPB
ii) AP. PB = CP . DP
Proof: i) In ∆ APC and ∆ DPB
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 16

Question 8.
In Fig. 2.62, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 17
Answer:
Given: In figure, two chords AB and CD of a circle intersect each other at the point P. (when produced) outside the circle.
To prove: i) ∆ PAC ~ ∆ PDB
ii )PA. PB = PC . PD
Proof: i) we know that in a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
Therefore, for cyclic quadrilateral ABCD.
consider ABCD ∆ PAC and ∆ PBD.
∠PAC = ∠PDB → (i)
∠PCA = ∠PBD → (2)
∆ PAC ~ ∆ PDB [A A similarity criterion]

ii) A PAC ~ ∆ PDB (Proved)
\(\frac{P A}{P D}=\frac{P C}{P B}\) [Corresponding sides of the similar ∆le are proportional]
PA . PB = PC . PD

KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6

Question 9.
In Fig. 2.63, D is a point on side BC of ∆ ABC such that \(\frac{B D}{C D}=\frac{A B}{A C}\) Prove that AD is the bisector of ∠BAC.

KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 18
Answer:
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 19
Given: In figure, D is a point on side BC. of ∆ ABC such that \(\frac{B D}{C D}=\frac{A B}{A C}\)
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 20

Question 10.
Nazima is fly fishing in a stream. The tip of her fishing rod ¡s 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 nad 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 Fig. 2.64)?
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 21
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?
Answer:
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 22
AC = 3m
Hence, she has 3 m string out.
Length of the string pulled in 12 seconds
at the rate of5 cm/ sec 5 × 12cm = 60 cm = O.6 m.
∴ Length of remaining string left out
= AD = 3.0 – 0.6 = 2.4m
KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6 23

In right angled ∆ ABD ∠B = 90°
AD2 = AB2 + BD2
BD2 = AD2 – AB2
[pythagoras theorem]
= (2.4)2 – (1.8)2
= 5.76 – 3.24 = 2.52
BD = 12.52 = 1.59 m (approx)
Hence, the horizontal distance of the fly form Nazirna after 12 seconds = 1.2 + 1.59 = 2.79 m.

KSEEB Solutions for Class 10 Maths Chapter 2 Triangles Ex 2.6

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