{"id":30932,"date":"2018-08-10T05:49:59","date_gmt":"2018-08-10T05:49:59","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=30932"},"modified":"2020-11-24T15:58:45","modified_gmt":"2020-11-24T10:28:45","slug":"ml-aggarwal-class-9-solutions-for-icse-maths-chapter-12-pythagoras-theorem","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/ml-aggarwal-class-9-solutions-for-icse-maths-chapter-12-pythagoras-theorem\/","title":{"rendered":"ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem"},"content":{"rendered":"

ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem<\/span><\/h2>\n

Question 1.<\/strong><\/span>
\nLengths 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:<\/strong>
\n(i) 3 cm, 8 cm, 6 cm<\/strong>
\n(ii) 13 cm, .12 cm, 5 cm<\/strong>
\n(iii) 1.4 cm, 4.8 cm, 5 cm<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 2.<\/strong><\/span>
\nFoot 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.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 3.<\/strong><\/span>
\nA 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?<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 4.<\/strong><\/span>
\nTwo 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.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 5.<\/strong><\/span>
\nIn a right-angled triangle, if hypotenuse is 20 cm and the ratio of the other two sides is 4:3, find the sides.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 6.<\/strong><\/span>
\nIf the sides of a triangle are in the ratio 3:4:5, prove that it is right-angled triangle.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 7.<\/strong><\/span>
\nFor going to a city B from city A, there is route via city C such that AC \u22a5 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.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 8.<\/strong><\/span>
\nThe 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.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 9.<\/strong><\/span>
\nABC is an isosceles triangle right angled at C. Prove that AB\u00b2 = 2AC\u00b2.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 10.<\/strong><\/span>
\nIn a triangle ABC, AD is perpendicular to BC. Prove that AB\u00b2 + CD\u00b2 = AC\u00b2 + BD\u00b2.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 11.<\/strong><\/span>
\nIn \u2206PQR, PD \u22a5 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).<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 12.<\/strong><\/span>
\nABC is an isosceles triangle with AB = AC = 12 cm and BC = 8 cm. Find the altitude on BC and Hence, calculate its area.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 13.<\/strong><\/span>
\nFind the area and the perimeter of a square whose diagonal is 10 cm long.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 14.<\/strong><\/span>
\n(a) In fig. (i) given below, ABCD is a quadrilateral in which AD = 13 cm, DC = 12 cm, BC = 3 cm, \u2220 ABD = \u2220BCD = 90\u00b0. Calculate the length of AB.<\/strong>
\n(b) In fig. (ii) given below, ABCD is a quadrilateral in which AB = AD, \u2220A = 90\u00b0 =\u2220C, BC = 8 cm and CD = 6 cm. Find AB and calculate the area of \u2206 ABD.<\/strong>
\n\"ML
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 15.<\/strong><\/span>
\n(a) In figure (i) given below, AB = 12 cm, AC = 13 cm, CE = 10 cm and DE = 6 cm.Calculate the length of BD.<\/strong>
\n(b) In figure (ii) given below, \u2220PSR = 90\u00b0, PQ = 10 cm, QS = 6 cm and RQ = 9 cm. Calculate the length of PR.<\/strong>
\n(c) In figure (iii) given below, \u2220 D = 90\u00b0, AB = 16 cm, BC = 12 cm and CA = 6 cm. Find CD.<\/strong>
\n\"ML
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 16.<\/strong><\/span>
\n(a) In figure (i) given below, BC = 5 cm,<\/strong>
\n\u2220B =90\u00b0, AB = 5AE, CD = 2AE and AC = ED. Calculate the lengths of EA, CD, AB and AC.<\/strong>
\n(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 = 4AD\u00b2 – 3AC\u00b2.<\/strong>
\n\"ML
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 17.<\/strong><\/span>
\nIn \u2206 ABC, AB = AC = x, BC = 10 cm and the area of \u2206 ABC is 60 cm\u00b2. Find x.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 18.<\/strong><\/span>
\nIn a rhombus, If diagonals are 30 cm and 40 cm, find its perimeter.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 19.<\/strong><\/span>
\n(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.<\/strong>
\n(b) In figure (ii) given below, AB || DC, \u2220 A = 90\u00b0, DC = 7 cm, AB = 17 cm and AC = 25 cm. Calculate BC.<\/strong>
\n(c) In figure (iii) given below, ABCD is a square of side 7 cm. if<\/strong>
\nAE = FC = CG = HA = 3 cm,<\/strong>
\n(i) prove that EFGH is a rectangle.<\/strong>
\n(ii) find the area and perimeter of EFGH.<\/strong>
\n\"ML
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML
\n\"ML
\n\"ML<\/p>\n

Question 20.<\/strong><\/span>
\nAD is perpendicular to the side BC of an equilateral \u0394 ABC. Prove that 4AD\u00b2 = 3AB\u00b2.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 21.<\/strong><\/span>
\nIn figure (i) given below, D and E are mid-points of the sides BC and CA respectively of a \u0394ABC, right angled at C.<\/strong>
\n\"ML
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 22.<\/strong><\/span>
\nIf AD, BE and CF are medians of \u0395ABC, prove that 3(AB\u00b2 + BC\u00b2 + CA\u00b2) = 4(AD\u00b2 + BE\u00b2 + CF\u00b2).<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML
\n\"ML
\n\"ML<\/p>\n

Question 23.<\/strong><\/span>
\n(a) In fig. (i) given below, the diagonals AC and BD of a quadrilateral ABCD intersect at O, at right angles. Prove that<\/strong>
\nAB\u00b2 + CD\u00b2 = AD\u00b2 + BC\u00b2.<\/strong>
\n(b) In figure (ii) given below, OD\u22a5BC, OE \u22a5CA and OF \u22a5 AB. Prove that :<\/strong>
\n(i) OA\u00b2 + OB\u00b2 + OC\u00b2 = AF\u00b2 + BD\u00b2 + CE\u00b2 + OD\u00b2 + OE\u00b2 + OF\u00b2.<\/strong>
\n(ii) OAF\u00b2 + BD\u00b2 + CE\u00b2 = FB\u00b2 + DC\u00b2 + EA\u00b2.<\/strong>
\n\"ML
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML
\n\"ML<\/p>\n

Question 24.<\/strong><\/span>
\nIn a quadrilateral, ABCD\u2220B = 90\u00b0 = \u2220D. Prove that 2 AC\u00b2 – BC2 = AB\u00b2 + AD\u00b2 + DC\u00b2.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 25.<\/strong><\/span>
\nIn a \u2206 ABC, \u2220 A = 90\u00b0, CA = AB and D is a point on AB produced. Prove that :<\/strong>
\nDC\u00b2 – BD\u00b2 = 2AB. AD.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 26.<\/strong><\/span>
\nIn an isosceles triangle ABC, AB = AC and D is a point on BC produced. Prove that AD\u00b2 = AC\u00b2 + BD.CD.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question P.Q.<\/strong><\/span>
\n(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.<\/strong>
\n(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.<\/strong>
\n(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.<\/strong>
\n(i) Calculate the length DE.<\/strong>
\n(ii) Prove that \u2206 DEC ~ AEBF and Hence, compute EF and BF.<\/strong>
\n\"ML
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML
\n\"ML
\n\"ML<\/p>\n

Multiple Choice Questions<\/strong><\/span><\/p>\n

Choose the correct answer from the given four options (1 to 7):<\/strong>
\nQuestion 1.<\/strong><\/span>
\nIn a \u2206ABC, if AB = 6\u221a3 cm, BC = 6 cm and AC = 12 cm, then \u2220B is<\/strong>
\n(a) 120\u00b0<\/strong>
\n(b) 90\u00b0<\/strong>
\n(c) 60\u00b0<\/strong>
\n(d) 45\u00b0<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 2.<\/strong><\/span>
\nIf the sides of a rectangular plot are 15 m and 8 m, then the length of its diagonal is<\/strong>
\n(a) 17 m<\/strong>
\n(b) 23 m<\/strong>
\n(c) 21 m<\/strong>
\n(d) 17 cm<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 3.<\/strong><\/span>
\nThe lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of the side of the rhombus is<\/strong>
\n(a) 9 cm<\/strong>
\n(b) 10 cm<\/strong>
\n(c) 8 cm<\/strong>
\n(d) 20 cm<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 4.<\/strong><\/span>
\nIf a side of a rhombus is 10 cm and one of the diagonals is 16 cm, then the length of the other diagonals is<\/strong>
\n(a) 6 cm<\/strong>
\n(b) 12 cm<\/strong>
\n(c) 20 cm<\/strong>
\n(d) 12 cm<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 5.<\/strong><\/span>
\nIf 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<\/strong>
\n(a) 18 m<\/strong>
\n(b) 8 m<\/strong>
\n(c) 6 m<\/strong>
\n(d) 4 m<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 6.<\/strong><\/span>
\nA girl walks 200 m towards East and then she walks ISO m towards North. The distance of the girl from the starting point is<\/strong>
\n(a) 350 m<\/strong>
\n(b) 250 m<\/strong>
\n(c) 300 m<\/strong>
\n(d) 225 m<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 7.<\/strong><\/span>
\nA 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<\/strong>
\n(a) 30 m<\/strong>
\n(b) 24 m<\/strong>
\n(c) 21 m<\/strong>
\n(d) 18 m<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Chapter Test<\/strong><\/span><\/p>\n

Question 1.<\/strong><\/span>
\n(a) In fig. (i) given below, AD \u22a5 BC, AB = 25 cm, AC = 17 cm and AD = 15 cm. Find the length of BC.<\/strong>
\n(b) In figure (ii) given below, \u2220BAC = 90\u00b0, \u2220ADC = 90\u00b0, AD = 6 cm, CD = 8 cm and BC = 26 cm. Find :<\/strong>
\n(i) AC (ii) AB (iii) area of the shaded region.<\/strong>
\n(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<\/strong>
\n(i) the length of BC (ii) the area of \u2206 ADE.<\/strong>
\n\"ML
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML
\n\"ML
\n\"ML<\/p>\n

Question 2.<\/strong><\/span>
\nIf in \u2206 ABC, AB > AC and ADI BC, prove that AB\u00b2 – AC\u00b2 = BD\u00b2 – CD\u00b2.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n

Question 3.<\/strong><\/span>
\nIn 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<\/strong>
\n(i) 9AQ\u00b2 = 9AC\u00b2 + 4BC\u00b2<\/strong>
\n(ii) 9BP\u00b2 = 9BC\u00b2 + 4AC\u00b2<\/strong>
\n(iii) 9(AQ\u00b2 + BP\u00b2) = 13AB\u00b2.<\/strong>
\nSolution:<\/strong><\/span>
\nA right angled \u2206 ABC in which \u2220 C
\n\"ML
\n\"ML<\/p>\n

Question 4.<\/strong><\/span>
\nIn the given figure, \u2206PQR is right angled at Q and points S and T trisect side QR. Prove that 8PT\u00b2 – 3PR\u00b2 + 5PS\u00b2.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 5.<\/strong><\/span>
\nIn a quadrilateral ABCD, \u2220B = 90\u00b0. If AD\u00b2 = AB\u00b2 + BC\u00b2 + CD\u00b2, prove that \u2220ACD = 90\u00b0.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 6.<\/strong><\/span>
\nIn the given figure, find the length of AD in terms of b and c.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML
\n\"ML<\/p>\n

Question 7.<\/strong><\/span>
\nABCD is a square, F is mid-point of AB and BE is one-third of BC. If area of \u2206FBE is 108 cm\u00b2, find the length of AC.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML
\n\"ML<\/p>\n

Question 8.<\/strong><\/span>
\nIn a triangle ABC, AB = AC and D is a point on side AC such that BC\u00b2<\/strong>\u00a0= AC x CD, Prove that BD = BC.<\/strong>
\nSolution:<\/strong><\/span>
\n\"ML<\/p>\n","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[3034],"tags":[6778,6777,6779,6775,6774,6782,6781,6776,6780],"yoast_head":"\nML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/ml-aggarwal-class-9-solutions-for-icse-maths-chapter-12-pythagoras-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 12 Pythagoras Theorem\" \/>\n<meta property=\"og:description\" content=\"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. 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