Misc 2 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry

Transcript

Misc 2 Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0). Let Δ ABC where AD, BE, CF are medians Since median bisects the opposite side D is Midpoint of BC E is Midpoint of AC F Midpoint of AB Calculating AD A (0, 0, 6) D (3, – 2, 0) AD = ﷐﷮﷐x2−x1﷯2+﷐y2−y1﷯2+﷐z2 −z1﷯2﷯ Hence x1 = 0, y1 = 0, z1 = 6 x2 = 3, y2 = – 2, z2 = 0 AD = ﷐﷮﷐3−0)2+(2−0﷯2+﷐0−6﷯2﷯ = ﷐﷮﷐3)2+(2﷯2+﷐−6﷯2﷯ = ﷐﷮9+4+36﷯ = ﷐﷮49﷯ = 7 Lenth of median AD = 7 Calculating BE B (0, 4, 0) E (3, 0, 3) BE = ﷐﷮﷐x2−x1﷯2+﷐y2−y1﷯2+﷐z2 −z1﷯2﷯ Hence x1 = 0, y1 = 4, z1 = 0 x2 = 3, y2 = 0, z2 = 3 BE = ﷐﷮﷐3−0)2+(0−4﷯2+﷐3−0﷯2﷯ = ﷐﷮﷐3)2+(−4﷯2+﷐3﷯2﷯ = ﷐﷮9+16+9﷯ = ﷐﷮34﷯ Length of median BE = ﷐﷮34﷯ Calculate CF C (6, 0, 0) F (0, 2, 3) CF = ﷐﷮﷐x2−x1﷯2+﷐y2−y1﷯2+﷐z2 −z1﷯2﷯ Hence x1 = 6, y1 = 0, z1 = 0 x2 = 0, y2 = 2, z2 = 3 CF = ﷐﷮﷐0−6)2+(2−0﷯2+﷐3−0﷯2﷯ = ﷐﷮﷐−6)2+(2﷯2+﷐3﷯2﷯ = ﷐﷮36+4+9﷯ = ﷐﷮49﷯ = 7 The Length of median CF = 7