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
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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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