with the help of distance formula,show that the points p(a,b+c),q(b,c+a) and r(c,a+b) are collinear points
Answers
Given : points P(a, b + c), Q (b, c + a) and R (c, a + b)
To Find : Show that points are collinear with the help of distance formula
Solution:
P(a, b + c),
Q (b, c + a)
R (c, a + b)
Distance between P & Q = √(b - a)² + (c + a - (b + c))²
= √(b - a)² + (a -b)²
= √(b - a)² + (-(b -a))²
= √2(b - a)²
= √2 .(b - a)
Distance between P & R = √(c - a)² + (a + b - (b + c))²
= √(c - a)² + (a -c)²
= √(c - a)² + (-(c -a))²
= √2(c - a)²
= √2 . (c - a)
Distance between Q & R = √(c - b)² + (a + b - (a + c))²
= √(c - b)² + (b -c)²
= √(c - b)² + (-(c -b))²
= √2(c - b)²
= √2 . (c - b)
PQ + QR = √2 .(b - a) + √2 . (c - b)
= √2 . (c - a)
= PR
PQ + PR = QR
Hence points are collinear
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