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MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315 Exercise 6.4 Q: 1 Let ABC ~ DEF their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC. . ic at m he 5 at 31 .m 44 w 1 w 0 w 81 09 Answer Q: 2 Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD. Answer Since AB || CD OCD (Alternate interior angles) OBA = ODC (Alternate interior angles) AOB = COD (Vertically opposite angles) Therefore AOB ~ COD in OAB = (By AAA rule) CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA, DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in south and west delhi EMAIL:1 [email protected] web site www.mathematic.in MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315 . ic at m he 5 at 31 .m 44 w 1 w 0 w 81 09 Q: 3 In figure 6.44, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that Answer Since ABC and DBC are one same base, Therefore ratio between their areas will be as ratio of their heights. Let us draw two perpendiculars AP and DM on line BC. APO and DMO, in In APO = DMO = 90° AOP = DOM (vertically opposite angles) OAP = ODM (remaining angle) Therefore APO ~ DMO (By AAA rule) CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA, DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in south and west delhi EMAIL:2 [email protected] web site www.mathematic.in MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315 . ic at m he 5 at 31 .m 44 w 1 w 0 w 81 09 Q: 4 If the areas of two similar triangles are equal, prove that they are congruent. Answer Let us assume two similar triangles as ABC ~ PQR Q: 5 D, E and F are respectively the mid-points of sides AB, BC and CA of DEF and ABC. Find the ratio of the area of ABC. Answer in Since D and E are mid points of ABC CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA, DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in south and west delhi EMAIL:3 [email protected] web site www.mathematic.in MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315 . ic at m he 5 at 31 .m 44 w 1 w 0 w 81 09 Q: 6 Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. Answer ABC ~ in Let us assume two similar triangles as PQR. Let AD and PS be the medians of these triangles. A = P, B = Q, C = R Since, AD and PS are medians CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA, DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in south and west delhi EMAIL:4 [email protected] web site www.mathematic.in MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315 . ic at m he 5 at 31 .m 44 w 1 w 0 w 81 09 Q: 7 Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals. Answer in Let ABCD be a square of side a. Therefore its diagonal Two desired equilateral triangles are formed as Side of an equilateral triangle ABE and DBF ABE described on one of its side = a CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA, DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in south and west delhi EMAIL:5 [email protected] web site www.mathematic.in MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315 Side of an equilateral triangle DBF described on one of its diagonal We know that equilateral triangles are having all its angles as 60º and all its sides of same length. So, all equilateral triangles are similar to each other. So, ratio between areas of these triangles will be equal to the square of the ratio between sides of these triangles. . ic at m he 5 at 31 .m 44 w 1 w 0 w 81 09 Q: 7 Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals. Answer Let ABCD be a square of side a. Therefore its diagonal Two desired equilateral triangles are formed as Side of an equilateral triangle ABE and DBF ABE described on one of its side = a Side of an equilateral triangle DBF described on one of its diagonal We know that equilateral triangles are having all its angles as 60º and all its sides of same length. So, all equilateral triangles are similar to each other. So, ratio between areas of these triangles will be equal to the square of the ratio between sides of these triangles. in Q: 8 ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the area of triangles ABC and BDE is (A) 2 : 1 (B) 1 : 2 (C) 4 : 1 CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA, DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in south and west delhi EMAIL:6 [email protected] web site www.mathematic.in MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315 (D) 1 : 4 Answer . ic at m he 5 at 31 .m 44 w 1 w 0 w 81 09 We know that equilateral triangles are having all its angles as 60º and all its sides of same length. So, all equilateral triangles are similar to each other. So, ratio between areas of these triangles will be equal to the square of the ratio between sides of these triangles. Let side of ABC = x Hence, (c) Q: 9 Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio (A) (B) (C) (D) 2:3 4:9 81 : 16 16 : 81 in Answer If, two triangles are similar to each other, ratio between areas of these triangles will be equal to the square of the ratio between sides of these triangles. Given that sides are in the ratio 4:9. Hence, (d). CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA, DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in south and west delhi EMAIL:7 [email protected] web site www.mathematic.in MATHEMATIC CENTER D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315 in . ic at m he 5 at 31 .m 44 w 1 w 0 w 81 09 CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA, DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in south and west delhi EMAIL:8 [email protected] web site www.mathematic.in