First slide

Introduction to 3-D Geometry

Question

$If a △ A B C the mid points of the sides A B, B C, C A are respectively (l, 0, 0), (0, m, 0) and (0, 0, n)$ $then$
$\frac{A B^{2} + B C^{2} + C A^{2}}{l^{2} + m^{2} + n^{2}} =$

Moderate

A

2
B

4
C

8
D

16

Solution

We have
$A + B = (2 l, 0, 0), B + C = (0, 2 m, 0) a n d C + A = (0, 0, 2 n) \Rightarrow A = (l, - m, n), B = (l, m, - n) a n d C = (- l, m, n) ∴ \frac{A B^{2} + B C^{2} + C A^{2}}{l^{2} + m^{2} + n^{2}} = \frac{4 m^{2} + 4 n^{2} + 4 l^{2} + 4 n^{2} + 4 l^{2} + 4 m^{2}}{l^{2} + m^{2} + n^{2}} = 8$

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