Question:
Using the distance formula, show that the points A(3, -2), B(5, 2) and C(8,8) are collinear.
Solution:
Given: The 3 points are A(3, -2), B(5, 2) and C(8, 8).
$A B=\sqrt{(5-3)^{2}+(2+2)^{2}}$
$=\sqrt{4+16}$
$=2 \sqrt{5}$ units
$B C=\sqrt{(8-5)^{2}+(8-2)^{2}}$
$=\sqrt{9+36}$
$=3 \sqrt{5}$ units
$A C=\sqrt{(8-3)^{2}+(8+2)^{2}}$
$=\sqrt{25+100}$
$=5 \sqrt{5}$ units …..(3)
From equations 1, 2 and 3, we have
⇒ AC = AB + BC
This is possible only if the points are collinear.
Therefore, the points A, B and C are collinear.
Hence, proved.