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Question
In ΔABC the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, `("AB"^2 + "BC"^2 + "CA"^2)/("l"^2 + "m"^2 + "n"^2)` is equal to ______.
Options
2
4
8
16
Solution
In ΔABC the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, `("AB"^2 + "BC"^2 + "CA"^2)/("l"^2 + "m"^2 + "n"^2)` is equal to 8.
Explanation:
From the figure
x1 + x2 = 2l, y1 + y2 = 0, z1 + z2 = 0
x2 + x3 = 0, y2 + y3 = 2m, z1 + z3 = 0
and x1 + x3 = 0, y1 + y3 = 0, z1 + z3 = 2n
On solving, we get x1 = l, x2 = l, x3 = –1
y1 = –m, y2 = m, y3 = m and z1 = n, z2 = –n, z3 = n
∴ Coordinates are A(l, –m, n), B(l, m, –n) and C(–l, m, n)
∴ `("AB"^2 + "BC"^2 + "CA"^2)/("l"^2 + "m"^2 + "n"^2)`
= `((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
