Class 10: Maths Chapter 2 solutions. Complete Class 10 Maths Chapter 2 Notes.

## Maharashtra Board Solutions for Class 10-Maths (Part 2): Chapter 2- Pythagoras Theorem

Maharashtra Board 10th Maths Chapter 2, Class 10 Maths Chapter 2 solutions

#### Practice Set 2.1

Question 1.
Identify, with reason, which of the following are Pythagorean triplets.

i. (3,5,4)
ii. (4,9,12)
iii. (5,12,13)
iv. (24,70,74)
v. (10,24,27)
vi. (11,60,61)
Solution:
i. Here, 52 = 25
32 + 42 = 9 + 16 = 25
∴ 52 = 32 + 42
The square of the largest number is equal to the sum of the squares of the other two numbers.
∴ (3,5,4) is a Pythagorean triplet.

ii. Here, 122 = 144
42 + 92= 16 + 81 =97
∴ 122 ≠ 42 + 92
The square of the largest number is not equal to the sum of the squares of the other two numbers.
∴ (4,9,12) is not a Pythagorean triplet.

iii. Here, 132 = 169
52 + 122 = 25 + 144 = 169
∴ 132 = 52 + 122
The square of the largest number is equal to the sum of the squares of the other two numbers.
∴ (5,12,13) is a Pythagorean triplet.

iv. Here, 742 = 5476
242 + 702 = 576 + 4900 = 5476
∴ 742 = 242 + 702
The square of the largest number is equal to the sum of the squares of the other two numbers.
∴ (24, 70,74) is a Pythagorean triplet.

v. Here, 272 = 729
102 + 242 = 100 + 576 = 676
∴ 272 ≠ 102 + 242
The square of the largest number is not equal to the sum of the squares of the other two numbers.
∴ (10,24,27) is not a Pythagorean triplet.

vi. Here, 612 = 3721
112 + 602 = 121 + 3600 = 3721
∴ 612 = 112 + 602
The square of the largest number is equal to the sum of the squares of the other two numbers.
∴ (11,60,61) is a Pythagorean triplet.

Question 2.
In the adjoining figure, ∠MNP = 90°, seg NQ ⊥ seg MP,MQ = 9, QP = 4, find NQ.

Solution:

In ∆MNP, ∠MNP = 90° and [Given]
seg NQ ⊥ seg MP
NQ2 = MQ × QP [Theorem of geometric mean]
∴ NQ = MQ×QP−−−−−−−−−√ [Taking square root of both sides]
= 9×4−−−−√
= 3 × 2
∴NQ = 6 units

Question 3.
In the adjoining figure, ∠QPR = 90°, seg PM ⊥ seg QR and Q – M – R, PM = 10, QM = 8, find QR.

Solution:

In ∆PQR, ∠QPR = 90° and [Given]
seg PM ⊥ seg QR
∴ PM2 = OM × MR [Theorem of geometric mean]
∴ 102 = 8 × MR
∴ MR = 1008
= 12.5
Now, QR = QM + MR [Q – M – R]
= 8 + 12.5
∴ QR = 20.5 units

Question 4.
See adjoining figure. Find RP and PS using the information given in ∆PSR.

Solution:

In ∆PSR, ∠S = 90°, ∠P = 30° [Given]
∴ ∠R = 60° [Remaining angle of a triangle]
∴ ∆PSR is a 30° – 60° – 90° triangle.
RS = 12 RP [Side opposite to 30°]
∴6 = 12 RP
∴ RP = 6 × 2 = 12 units
Also, PS = 3√2 RP [Side opposite to 60°]
= 3√2 × 12
= 63–√ units
∴ RP = 12 units, PS = 6 3–√ units

Question 5.
For finding AB and BC with the help of information given in the adjoining figure, complete the following activity.

Solution:

AB = BC [Given]
∴ ∠BAC = ∠BCA [Isosceles triangle theorem]
Let ∠BAC = ∠BCA = x (i)
In ∆ABC, ∠A + ∠B + ∠C = 180° [Sum of the measures of the angles of a triangle is 180°]
∴ x + 90° + x = 180° [From (i)]
∴ 2x = 90°
∴ x = 90°2 [From (i)]
∴ x = 45°

Question 6.
Find the side and perimeter of a square whose diagonal is 10 cm.

Solution:

Let ꠸ABCD be the given square.
l(diagonal AC) = 10 cm
Let the side of the square be ‘x’ cm.
In ∆ABC,
∠B = 90° [Angle of a square]
∴ AC2 = AB2 + BC2 [Pythagoras theorem]
∴ 102 = x2 + x2
∴ 100 = 2x2
∴ x2 = 1002
∴x2 = 50
∴ x = 50−−√ [Taking square root of both sides]
= =25×2−−−−−√=52–√
∴side of square is 52–√ cm.
= 4 × 5 2–√
∴ Perimeter of square = 20 2–√ cm

Question 7.
In the adjoining figure, ∠DFE = 90°, FG ⊥ ED. If GD = 8, FG = 12, find

i. EG
ii. FD, and
iii. EF

Solution:
i. In ∆DEF, ∠DFE = 90° and FG ⊥ ED [Given]
∴ FG2 = GD × EG [Theorem of geometric mean]
∴ 122 = 8 × EG .
∴ EG = 1448
∴ EG = 18 units

ii. In ∆FGD, ∠FGD = 90° [Given]
∴ FD2 = FG2 + GD2 [Pythagoras theorem]
= 122 + 82 = 144 + 64
= 208
∴ FD = 208−−−√ [Taking square root of both sides]
∴ FD = 4 13−−√ units

iii. In ∆EGF, ∠EGF = 90° [Given]
∴ EF2 = EG2 + FG2 [Pythagoras theorem]
= 182 + 122 = 324 + 144
= 468
∴ EF = 468−−−√ [Taking square root of both sides]
∴ EF = 6 13−−√ units

Question 8.
Find the diagonal of a rectangle whose length is 35 cm and breadth is 12 cm.

Solution:

Let ꠸ABCD be the given rectangle.
AB = 12 cm, BC 35 cm
In ∆ABC, ∠B = 90° [Angle of a rectangle]
∴ AC2 = AB2 + BC2 [Pythagoras theorem]
= 122 + 352
= 144 + 1225
= 1369
∴ AC = 1369−−−−√ [Taking square root of both sides]
= 37 cm
∴ The diagonal of the rectangle is 37 cm.

Question 9.
In the adjoining figure, M is the midpoint of QR. ∠PRQ = 90°.
Prove that, PQ2 = 4 PM2 – 3 PR2.

Solution:

Proof:
In ∆PQR, ∠PRQ = 90° [Given]
PQ2 = PR2 + QR2 (i) [Pythagoras theorem]
RM = 12 QR [M is the midpoint of QR]
∴ 2RM = QR (ii)
∴ PQ2 = PR2 + (2RM)2 [From (i) and (ii)]
∴ PQ2 = PR2 + 4RM2 (iii)
Now, in ∆PRM, ∠PRM = 90° [Given]
∴ PM2 = PR2 + RM2 [Pythagoras theorem]
∴ RM2 = PM2 – PR2 (iv)
∴ PQ2 = PR2 + 4 (PM2 – PR2) [From (iii) and (iv)]
∴ PQ2 = PR2 + 4 PM2 – 4 PR2
∴ PQ2 = 4 PM2 – 3 PR2

Question 10.
Walls of two buildings on either side of a street are parallel to each other. A ladder 5.8 m long is placed on the street such that its top just reaches the window of a building at the height of 4 m. On turning the ladder over to the other side of the street, its top touches the window of the other building at a height 4.2 m. Find the width of the street.
Solution:

Let AC and CE represent the ladder of length 5.8 m, and A and E represent windows of the buildings on the opposite sides of the street. BD is the width of the street.

AB = 4 m and ED = 4.2 m
In ∆ABC, ∠B = 90° [Given]
AC2 = AB2 + BC2 [Pythagoras theorem]
∴ 5.82 = 42 + BC2
∴ 5.82 – 42 = BC2
∴ (5.8 – 4) (5.8 + 4) = BC2
∴ 1.8 × 9.8 = BC2

CE2 = CD2 + DE2 [Pythagoras theorem]
∴ 5.82 = CD2 + 4.22
∴ 5.82 – 4.22 = CD2
∴ (5.8 – 4.2) (5.8 + 4.2) = CD2
∴ 1.6 × 10 = CD2
∴ CD2 = 16
∴ CD = 4m (ii) [Taking square root of both sides]
Now, BD = BC + CD [B – C – D]
= 4.2 + 4 [From (i) and (ii)]
= 8.2 m
∴ The width of the street is 8.2 metres.

Question 1.
Verify that (3,4,5), (5,12,13), (8,15,17), (24,25,7) are Pythagorean triplets. (Textbook pg. no. 30)
Solution:

i. Here, 52 = 25
32 + 42 = 9 + 16 = 25
∴ 52 = 32 + 42
The square of the largest number is equal to the sum of the squares of the other two numbers.
∴ 3,4,5 is a Pythagorean triplet.

ii. Here, 132 = 169
52 + 122 = 25 + 144 = 169
∴ 132 = 52 + 122
The square of the largest number is equal to the sum of the squares of the other two numbers.
∴ 5,12,13 is a Pythagorean triplet.

iii. Here, 172 = 289
82 + 152 = 64 + 225 = 289
∴ 172 = 82 + 152
The square of the largest number is equal to the sum of the squares of the other two numbers.
∴ 8,15,17 is a Pythagorean triplet.

iv. Here, 252 = 625
72 + 24= 49 + 576 = 625
∴ 252 = 72 + 242
The square of the largest number is equal to the sum of the squares of the other two numbers.
∴ 24,25, 7 is a Pythagorean triplet.

Question 2.
Assign different values to a and b and obtain 5 Pythagorean triplets. (Textbook pg. no. 31)
Solution:

i. Let a = 2, b = 1
a2 + b2 = 22 + 12 = 4 + 1 = 5
a2 – b2 = 22 – 12 = 4 – 1 = 3
2ab = 2 × 2 × 1 = 4
∴ (5, 3, 4) is a Pythagorean triplet.

ii. Let a = 4,b = 3
a2 + b2 = 42 + 32 = 16 + 9 = 25
a2 – b2 = 42 – 32 = 16 – 9 = 7
2ab = 2 × 4 × 3 = 24
∴ (25, 7, 24) is a Pythagorean triplet.

iii. Let a = 5, b = 2
a2 + b2 = 52 + 22 = 25 + 4 = 29
a2 – b2 = 52 – 22 = 25 – 4 = 21
2ab = 2 × 5 × 2 = 20
∴ (29, 21, 20) is a Pythagorean triplet.

iv. Let a = 4,b = 1
a2 + b2 = 42 + 12 = 16 + 1 = 17
a2 – b2 = 42 – 12 = 16 – 1 = 15
2ab = 2 × 4 × 1 = 8
∴ (17, 15, 8) is a Pythagorean triplet.

v. Let a = 9, b = 7
a2 + b2 = 92 + 72 = 81 + 49 = 130
a2 – b2 = 92 – 72 = 81 – 49 = 32
2ab = 2 × 9 × 7 = 126
∴ (130,32,126) is a Pythagorean triplet.

Note: Numbers in Pythagorean triplet can be written in any order.

#### Practice Set 2.2

Question 1.
In ∆PQR, point S is the midpoint of side QR. If PQ = 11, PR = 17, PS = 13, find QR.

Solution:
In ∆PQR, point S is the midpoint of side QR. [Given]
∴ seg PS is the median.
∴ PQ2 + PR2 = 2 PS2 + 2 SR2 [Apollonius theorem]
∴ 112 + 172 = 2 (13)2 + 2 SR2
∴ 121 + 289 = 2 (169)+ 2 SR2
∴ 410 = 338+ 2 SR2
∴ 2 SR2 = 410 – 338
∴ 2 SR2 = 72
∴ SR2 = 722 = 36
∴ SR = 36−−√ [Taking square root of both sides]
= 6 units Now, QR = 2 SR [S is the midpoint of QR]
= 2 × 6
∴ QR = 12 units

Question 2.
In ∆ABC, AB = 10, AC = 7, BC = 9, then find the length of the median drawn from point C to side AB.
Solution:

Let CD be the median drawn from the vertex C to side AB.
BD = 12 AB [D is the midpoint of AB]
= 12 × 10 = 5 units

In ∆ABC, seg CD is the median. [Given]
∴ AC2 + BC2 = 2 CD2 + 2 BD2 [Apollonius theorem]
∴ 72 + 92 = 2 CD2 + 2 (5)2
∴ 49 + 81 = 2 CD2 + 2 (25)
∴ 130 = 2 CD2 + 50
∴ 2 CD2 = 130 – 50
∴ 2 CD2 = 80
∴ CD2 = 802 = 40
∴ CD = 40−−√ [Taking square root of both sides]
= 2 10−−√ units
∴ The length of the median drawn from point C to side AB is 2 10−−√ units.

Question 3.
In the adjoining figure, seg PS is the median of APQR and PT ⊥ QR. Prove that,

i. PR2 = PS2 + QR × ST + (QR2)2
ii. PQ2 = PS2 – QR × ST + (QR2)2
Solution:
i. QS = SR = 12 QR (i) [S is the midpoint of side QR]

∴ In ∆PSR, ∠PSR is an obtuse angle [Given]
and PT ⊥ SR [Given, Q-S-R]
∴ PR2 = SR2 +PS2 + 2 SR × ST (ii) [Application of Pythagoras theorem]
∴ PR2 = (12 QR)2 + PS2 + 2 (12 QR) × ST [From (i) and (ii)]
∴ PR2 = (QR2)2 + PS2 + QR × ST
∴ PR2 = PS2 + QR × ST + (QR2)2

ii. In.∆PQS, ∠PSQ is an acute angle and [Given]

PT ⊥QS [Given, Q-S-R]
∴ PQ2 = QS2 + PS2 – 2 QS × ST (iii) [Application of Pythagoras theorem]
∴ PR2 = (12 QR)2 + PS2 – 2 (12 QR) × ST [From (i) and (iii)]
∴ PR2 = (QR2)2 + PS2 – QR × ST
∴ PR2 = PS2 – QR × ST + (QR2)2

Question 4.
In ∆ABC, point M is the midpoint of side BC. If AB+ AC2 = 290 cm, AM = 8 cm, find BC.

Solution:

In ∆ABC, point M is the midpoint of side BC. [Given]
∴ seg AM is the median.
∴ AB2 + AC2 = 2 AM2 + 2 MC2 [Apollonius theorem]
∴ 290 = 2 (8)2 + 2 MC2
∴ 145 = 64 + MC2 [Dividing both sides by 2]
∴ MC2 = 145 – 64
∴ MC2 = 81
∴ MC = 81−−√ [Taking square root of both sides]
MC = 9 cm
Now, BC = 2 MC [M is the midpoint of BC]
= 2 × 9
∴ BC = 18 cm

Question 5.
In the adjoining figure, point T is in the interior of rectangle PQRS. Prove that, TS2 + TQ2 = TP2 + TR2. (As shown in the figure, draw seg AB || side SR and A – T – B)

Given: ꠸PQRS is a rectangle.
Point T is in the interior of ꠸PQRS.
To prove: TS2 + TQ2 = TP2 + TR2
Construction: Draw seg AB || side SR such that A – T – B.
Solution:
Proof:
꠸PQRS is a rectangle. [Given]
∴ PS = QR (i) [Opposite sides of a rectangle]
In ꠸ASRB,
∠S = ∠R = 90° (ii) [Angles of rectangle PQRS]
side AB || side SR [Construction]
Also ∠A = ∠S = 90° [Interior angle theorem, from (ii)]
∠B = ∠R = 90°
∴ ∠A = ∠B = ∠S = ∠R = 90° (iii)
∴ ꠸ASRB is a rectangle.
∴ AS = BR (iv) [Opposite sides of a rectanglel

In ∆PTS, ∠PST is an acute angle
and seg AT ⊥ side PS [From (iii)]
∴ TP2 = PS2 + TS2 – 2 PS.AS (v) [Application of Pythagoras theorem]
In ∆TQR., ∠TRQ is an acute angle
and seg BT ⊥ side QR [From (iii)]
∴ TQ2 = RQ2 + TR2 – 2 RQ.BR (vi) [Application of pythagoras theorem]

TP2 – TQ2 = PS2 + TS2 – 2PS.AS
-RQ2 – TR2 + 2RQ.BR [Subtracting (vi) from (v)]
∴ TP2 – TQ2 = TS2 – TR2 + PS2
– RQ2 -2 PS.AS +2 RQ.BR
∴ TP2 – TQ2 = TS2 – TR2 + PS2
– PS2 – 2 PS.BR + 2PS.BR [From (i) and (iv)]
∴ TP2 – TQ2 = TS2 – TR2
∴ TS2 + TQ2 = TP2 + TR2

Question 1.
In ∆ABC, ∠C is an acute angle, seg AD Iseg BC. Prove that: AB2 = BC2 + A2 – 2 BC × DC. (Textbook pg. no. 44)
Given: ∠C is an acute angle, seg AD ⊥ seg BC.
To prove: AB2 = BC2 + AC2 – 2BC × DC
Solution:

Proof:
∴ LetAB = c, AC = b, AD = p,

∴ BC = a, DC = x
BD + DC = BC [B – D – C]
∴ BD = BC – DC
∴ BD = a – x
In ∆ABD, ∠D = 90° [Given]
AB2 = BD2 + AD2 [Pythagoras theorem]
∴ c2 = (a – x)2 + [P2] (i)
∴ c2 = a2 – 2ax + x2 + [P2]
In ∆ADC, ∠D = 90° [Given]
AC2 = AD2 + CD2 [Pythagoras theorem]
∴ b2 = p2 + [X2]
∴ p2 = b2 – [X2] (ii)
∴ c2 = a2 – 2ax + x2 + b2 – x2 [Substituting (ii) in (i)]
∴ c2 = a2 + b2 – 2ax
∴ AB2 = BC2 + AC2 – 2 BC × DC

Question 2.
In ∆ABC, ∠ACB is an obtuse angle, seg AD ⊥ seg BC. Prove that: AB2 = BC2 + AC2 + 2 BC × CD. (Textbook pg. no. 40 and 4.1)
Given: ∠ACB is an obtuse angle, seg AD ⊥ seg BC.
To prove: AB2 = BC2 + AC2 + 2BC × CD
Solution:

Proof:

Let AD = p, AC = b, AB = c,
BC = a, DC = x
BD = BC + DC [B – C – D]
∴ BD = a + x
In ∆ADB, ∠D = 90° [Given]
AB2 = BD2 + AD2 [Pythagoras theorem]
∴ c2 = (a + x)2 + p2 (i)
∴ c2 = a2 + 2ax + x2 + p2
Also, in ∆ADC, ∠D = 90° [Given]
AC2 = CD2 + AD2 [Pythagoras theorem]
∴ b2 = x2 + p2
∴ p2 = b2 – x(ii)
∴ c2 = a2 + 2ax + x2 + b2 – x2 [Substituting (ii) in (i)]
∴ c2 = a2 + b2 + 2ax
∴ AB2 = BC2 + AC2 + 2 BC × CD

Question 3.
In ∆ABC, if M is the midpoint of side BC and seg AM ⊥seg BC, then prove that
AB2 + AC2 = 2 AM2 + 2 BM2. (Textbook pg, no. 41)
Given: In ∆ABC, M is the midpoint of side BC and seg AM ⊥ seg BC.
To prove: AB2 + AC2 = 2 AM2 + 2 BM2
Solution:

Proof:

In ∆AMB, ∠M = 90° [segAM ⊥ segBC]
∴ AB2 = AM2 + BM2 (i) [Pythagoras theorem]
Also, in ∆AMC, ∠M = 90° [seg AM ⊥ seg BC]
∴ AC2 = AM2 + MC2 (ii) [Pythagoras theorem]
∴ AB2 + AC= AM2 + BM2 + AM2 + MC2 [Adding (i) and (ii)]
∴ AB2 + AC2 = 2 AM2 + BM2 + BM2 [∵ BM = MC (M is the midpoint of BC)]
∴ AB2 + AC2 = 2 AM2 + 2 BM2

Maharashtra Board Solutions for Class 10-Maths (Part 2): Chapter 2- Pythagoras Theorem

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