ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem

ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem

Question 1.
Lengths of sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse:
(i) 3 cm, 8 cm, 6 cm
(ii) 13 cm, .12 cm, 5 cm
(iii) 1.4 cm, 4.8 cm, 5 cm
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q1.1

Question 2.
Foot of a 10 m long ladder leaning against a vertical well is 6 m away from the base of the wail. Find the height of the point on the wall where the top of the ladder reaches.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q2.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q2.2

Question 3.
A guy attached a wire 24 m long to a vertical pole of height 18 m and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taught?
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q3.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q3.2

Question 4.
Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q4.1

Question 5.
In a right-angled triangle, if hypotenuse is 20 cm and the ratio of the other two sides is 4:3, find the sides.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q5.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q5.2

Question 6.
If the sides of a triangle are in the ratio 3:4:5, prove that it is right-angled triangle.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q6.1

Question 7.
For going to a city B from city A, there is route via city C such that AC ⊥ CB, AC = 2x km and CB=2(x+ 7) km. It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of highway.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q7.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q7.2

Question 8.
The hypotenuse of right triangle is 6m more than twice the shortest side. If the third side is 2m less than the hypotenuse, find the sides of the triangle.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q8.1

Question 9.
ABC is an isosceles triangle right angled at C. Prove that AB² = 2AC².
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q9.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q9.2

Question 10.
In a triangle ABC, AD is perpendicular to BC. Prove that AB² + CD² = AC² + BD².
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q10.1

Question 11.
In ∆PQR, PD ⊥ QR, such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, prove that (a + b) (a – b) = (c + d) (c – d).
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q11.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q11.2

Question 12.
ABC is an isosceles triangle with AB = AC = 12 cm and BC = 8 cm. Find the altitude on BC and Hence, calculate its area.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q12.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q12.2

Question 13.
Find the area and the perimeter of a square whose diagonal is 10 cm long.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q13.1

Question 14.
(a) In fig. (i) given below, ABCD is a quadrilateral in which AD = 13 cm, DC = 12 cm, BC = 3 cm, ∠ ABD = ∠BCD = 90°. Calculate the length of AB.
(b) In fig. (ii) given below, ABCD is a quadrilateral in which AB = AD, ∠A = 90° =∠C, BC = 8 cm and CD = 6 cm. Find AB and calculate the area of ∆ ABD.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q14.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q14.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q14.3

Question 15.
(a) In figure (i) given below, AB = 12 cm, AC = 13 cm, CE = 10 cm and DE = 6 cm.Calculate the length of BD.
(b) In figure (ii) given below, ∠PSR = 90°, PQ = 10 cm, QS = 6 cm and RQ = 9 cm. Calculate the length of PR.
(c) In figure (iii) given below, ∠ D = 90°, AB = 16 cm, BC = 12 cm and CA = 6 cm. Find CD.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q15.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q15.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q15.3

Question 16.
(a) In figure (i) given below, BC = 5 cm,
∠B =90°, AB = 5AE, CD = 2AE and AC = ED. Calculate the lengths of EA, CD, AB and AC.
(b) In the figure (ii) given below, ABC is a right triangle right angled at C. If D is mid-point of BC, prove that AB2 = 4AD² – 3AC².
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q16.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q16.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q16.3

Question 17.
In ∆ ABC, AB = AC = x, BC = 10 cm and the area of ∆ ABC is 60 cm². Find x.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q17.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q17.2

Question 18.
In a rhombus, If diagonals are 30 cm and 40 cm, find its perimeter.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q18.1

Question 19.
(a) In figure (i) given below, AB || DC, BC = AD = 13 cm. AB = 22 cm and DC = 12cm. Calculate the height of the trapezium ABCD.
(b) In figure (ii) given below, AB || DC, ∠ A = 90°, DC = 7 cm, AB = 17 cm and AC = 25 cm. Calculate BC.
(c) In figure (iii) given below, ABCD is a square of side 7 cm. if
AE = FC = CG = HA = 3 cm,
(i) prove that EFGH is a rectangle.
(ii) find the area and perimeter of EFGH.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q19.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q19.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q19.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q19.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q19.5

Question 20.
AD is perpendicular to the side BC of an equilateral Δ ABC. Prove that 4AD² = 3AB².
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q20.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q20.2

Question 21.
In figure (i) given below, D and E are mid-points of the sides BC and CA respectively of a ΔABC, right angled at C.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q21.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q21.2

Question 22.
If AD, BE and CF are medians of ΕABC, prove that 3(AB² + BC² + CA²) = 4(AD² + BE² + CF²).
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q22.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q22.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q22.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q22.4

Question 23.
(a) In fig. (i) given below, the diagonals AC and BD of a quadrilateral ABCD intersect at O, at right angles. Prove that
AB² + CD² = AD² + BC².
(b) In figure (ii) given below, OD⊥BC, OE ⊥CA and OF ⊥ AB. Prove that :
(i) OA² + OB² + OC² = AF² + BD² + CE² + OD² + OE² + OF².
(ii) OAF² + BD² + CE² = FB² + DC² + EA².
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q23.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q23.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q23.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q23.4

Question 24.
In a quadrilateral, ABCD∠B = 90° = ∠D. Prove that 2 AC² – BC2 = AB² + AD² + DC².
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q24.1

Question 25.
In a ∆ ABC, ∠ A = 90°, CA = AB and D is a point on AB produced. Prove that :
DC² – BD² = 2AB. AD.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q25.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q25.2

Question 26.
In an isosceles triangle ABC, AB = AC and D is a point on BC produced. Prove that AD² = AC² + BD.CD.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Q26.1

Question P.Q.
(a) In figure (i) given below, PQR is a right angled triangle, right angled at Q. XY is parallel to QR. PQ = 6 cm, PY = 4 cm and PX : OX = 1:2. Calculate the length of PR and QR.
(b) In figure (ii) given below, ABC is a right angled triangle, right angled at B.DE || BC.AB = 12 cm, AE = 5 cm and AD : DB = 1: 2. Calculate the perimeter of A ABC.
(c)In figure (iii) given below. ABCD is a rectangle, AB = 12 cm, BC – 8 cm and E is a point on BC such that CE = 5 cm. DE when produced meets AB produced at F.
(i) Calculate the length DE.
(ii) Prove that ∆ DEC ~ AEBF and Hence, compute EF and BF.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Qp1.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Qp1.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Qp1.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Qp1.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem Qp1.5

Multiple Choice Questions

Choose the correct answer from the given four options (1 to 7):
Question 1.
In a ∆ABC, if AB = 6√3 cm, BC = 6 cm and AC = 12 cm, then ∠B is
(a) 120°
(b) 90°
(c) 60°
(d) 45°
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem mul Q1.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem mul Q1.2

Question 2.
If the sides of a rectangular plot are 15 m and 8 m, then the length of its diagonal is
(a) 17 m
(b) 23 m
(c) 21 m
(d) 17 cm
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem mul Q2.1

Question 3.
The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of the side of the rhombus is
(a) 9 cm
(b) 10 cm
(c) 8 cm
(d) 20 cm
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem mul Q3.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem mul Q3.2

Question 4.
If a side of a rhombus is 10 cm and one of the diagonals is 16 cm, then the length of the other diagonals is
(a) 6 cm
(b) 12 cm
(c) 20 cm
(d) 12 cm
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem mul Q4.1

Question 5.
If a ladder 10 m long reaches a window 8 m above the ground, then the distance of the foot of the ladder from the base of the wall is
(a) 18 m
(b) 8 m
(c) 6 m
(d) 4 m
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem mul Q5.1

Question 6.
A girl walks 200 m towards East and then she walks ISO m towards North. The distance of the girl from the starting point is
(a) 350 m
(b) 250 m
(c) 300 m
(d) 225 m
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem mul Q6.1

Question 7.
A ladder reaches a window 12 m above the ground on one side of the street. Keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window 9 m high. If the length of the ladder is 15 m, then the width of the street is
(a) 30 m
(b) 24 m
(c) 21 m
(d) 18 m
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem mul Q7.1

Chapter Test

Question 1.
(a) In fig. (i) given below, AD ⊥ BC, AB = 25 cm, AC = 17 cm and AD = 15 cm. Find the length of BC.
(b) In figure (ii) given below, ∠BAC = 90°, ∠ADC = 90°, AD = 6 cm, CD = 8 cm and BC = 26 cm. Find :
(i) AC (ii) AB (iii) area of the shaded region.
(c) In figure (iii) given below, triangle ABC is right angled at B. Given that AB = 9 cm, AC = 15 cm and D, E are mid-points of the sides AB and AC respectively, calculate
(i) the length of BC (ii) the area of ∆ ADE.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q1.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q1.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q1.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q1.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q1.5

Question 2.
If in ∆ ABC, AB > AC and ADI BC, prove that AB² – AC² = BD² – CD².
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q2.1

Question 3.
In a right angled triangle ABC, right angled at C, P and Q are the points on the sides CA and CB respectively which divide these sides in the ratio 2:1. Prove that
(i) 9AQ² = 9AC² + 4BC²
(ii) 9BP² = 9BC² + 4AC²
(iii) 9(AQ² + BP²) = 13AB².
Solution:
A right angled ∆ ABC in which ∠ C
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q3.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q3.2

Question 4.
In the given figure, ∆PQR is right angled at Q and points S and T trisect side QR. Prove that 8PT² – 3PR² + 5PS².
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q4.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q4.2

Question 5.
In a quadrilateral ABCD, ∠B = 90°. If AD² = AB² + BC² + CD², prove that ∠ACD = 90°.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q5.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q5.2

Question 6.
In the given figure, find the length of AD in terms of b and c.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q6.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q6.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q6.3

Question 7.
ABCD is a square, F is mid-point of AB and BE is one-third of BC. If area of ∆FBE is 108 cm², find the length of AC.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q7.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q7.2

Question 8.
In a triangle ABC, AB = AC and D is a point on side AC such that BC² = AC x CD, Prove that BD = BC.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem ch Q8.1

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