# Example, 11 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry

Last updated at Dec. 24, 2019 by Teachoo

Last updated at Dec. 24, 2019 by Teachoo

Transcript

Example, 11 Show that the points A (1, 2, 3), B (–1, –2, –1), C (2, 3, 2) and D (4, 7, 6) are the vertices of a parallelogram ABCD, but it is not a rectangle. Difference between We need to prove that ABCD is a parallelogram but not a rectangle So, We need to prove that Opposite side are equal i.e. AB = CD and BC = DA but Diagonals Not equal ( AC ≠ BD) Calculating AB A (1, 2, 3) B ( – 1, – 2, – 1) AB = x2−x12+y2−y12+z2 −z12 x1 = 1, y1 = 2, z1 = 3 x2 = – 1, y2 = – 2, z2 = – 1 AB = −1−12+−2−22+−1−32 = −22+−42+−42 = 4+16+16 = 36 = 6 Calculating BC B ( – 1, – 2, – 1) C (2, 3, 2) BC = x2−x12+y2−y12+z2 −z12 x1 = – 1, y1 = – 2, z1 = – 1 x2 = 2, y2 = 3, z2 = 2 BC = 2−(−12+3−(−2)2+2−(−1)2 = 2+12+3+22+2+12 = 32+52+32 = 9+25+9 = 43 Calculating CD C (2, 3, 2) D (4, 7, 6) CD = x2−x12+y2−y12+z2 −z12 x2 = 2, y2 = 3, z2 = 2 x2 = 4, y2 = 7, z2 = 6 CD = 4−22+7−32+6−22 = 22+42+42 = 4+16+16 = 36 = 6 Calculating DA D (4, 7, 6) A (1, 2, 3) DA = x2−x12+y2−y12+z2 −z12 x2 = 4, y2 = 7, z2 = 6 x1 = 1, y1 = 2, z1 = 3 DA = 1−42+2−72+3−62 = −32+−42+−32 = 9+16+9 = 43 Hence AB = CD = 6 & BC = DA = 43 Thus, Opposite side are equal ∴ ABCD is Parallelogram Now we need to prove, diagonals are not equal i.e. AC ≠ BD Calculating AC A (1, 2, 3) C (2, 3, 2) AC = x2−x12+y2−y12+z2 −z12 Here, x2 = 1, y2 = 2, z2 = 3 x1 = 2, y1 = 3, z1 = 2 AC = 2−12+3−22+2−32 = 12+12+−12 = 1+1+1 = 3 Calculating BD B ( – 1, – 2, – 1) , D (4, 7, 6) BD = x2−x12+y2−y12+z2 −z12 Here, x1 = –1, y1 = –2, z1 = –1 x2 = 4, y2 = 7, z2 = 6 BD = x2−x12+y2−y12+z2 −z12 BD = 4−(−1)2+7−(−2)2+6−(−1)2 = 4+12+7+22+6+12 = 52+92+72 = 25+81+49 = 155 Now, AC = 3 & BD = 155 As 3 ≠ 155 Hence AC ≠ BD Since Diagonals Not equal Hence it is ABCD is not rectangle ∴ ABCD is a parallelogram but not a rectangle Hence proved

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.