Example 5 - Are A (3, 6, 9), B (10, 20, 30), C (25, -41, 5) - Examples

  1. Chapter 12 Class 11 Introduction to Three Dimensional Geometry
  2. Serial order wise

Transcript

Example 5 Are the points A (3, 6, 9), B (10, 20, 30) and C (25, 41, 5), the vertices of a right angled triangle? Lets first calculate distances AB, BC and AC and then apply Pythagoras theorem to determine whether it is right angle triangle Calculating AB A (3, 6, 9) B (10, 20, 30) AB = x2 x1 2+ y2 y1 2+ z2 z1 2 Here, x1 = 3, y1 = 6, z1 = 9 x2 = 10, y2 = 20, z2 = 30 AB = 10 3)2+(20 6 2+ 30 9 2 = 7)2+(14 2+ 21 2 = 49+196+441 = 686 Calculating BC B (10, 20, 30) C (25, 41, 5) BC = x2 x1 2+ y2 y1 2+ z2 z1 2 Here x1 = 10, y1 = 20, z1 = 30 x2 = 25, y2 = 41, z2 = 5 BC = 25 10)2+( 41 20 2+ 5 30 2 = 15)2+( 61 2+ 25 2 = 225+3721+625 = 4571 Calculating CA C (25, 41, 5) A (3, 6, 9) CA = x2 x1 2+ y2 y1 2+ z2 z1 2 x1 = 25, y1 = 14, z1 = 5 x2 = 3, y2 = 6, z2 = 9 AB = 3 25)2+(6 ( 41) 2+ 9 5 2 = 22)2+(6+41 2+ 4 2 = 484+ 47 2+16 = 484+2209+16 = 2709 Now AB = 686 , BC = 4571 , CA = 2709 In Right angle tringle, (Hypotenuse)2 = (Height)2 + (Base)2 Since 4571 is the biggest of the three sides , we take Hypotenuse = 4571 Hence we have to prove ( 4571 )2 = ( 686 )2 + ( 2709 )2 Thus, L.H.S R.H.S Hence, It is not a right angle triangle

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