# If the Origin is the Centroid of the Triangle Whose Vertices Are A(2, P, –3), B(Q, –2, 5) and R(–5, 1, R), Then Find the Values of P, Q, R. - Mathematics and Statistics

Sum

If the origin is the centroid of the triangle whose vertices are A(2, p, –3), B(q, –2, 5) and C(–5, 1, r), then find the values of p, q, r.

#### Solution

Let bara, barb,barc be the position vectors of triangle ABC whose vertices are A(2, p, –3), B(q, –2, 5) and C(–5, 1, r)

:.bara = 2hati + pbarj - 3bark, barb = qbari - 2barj + 5bark, barc = -5bari +  barj + rbark

Given that origin O is the centroid of triangle ABC

:. barO = (bara+barb+barc)/3

:. bara+barb+barc = barO

2hati + phatj - 3hatk + hatj - 2hatj +5hatk - 5hati + hatj + rhatk = barO

=> (2+q-5)hati + (p-2+1)hatj + (-3+5+r)hatk = 0hati + 0hatj + 0hatk

by equality of vectors

2 + q - 5 = 0 ⇒ q = 3

p - 2 + 1 = 0 ⇒ p = 1

-3 + 5 + r = 0 ⇒ r = -2

∴ p = 1, q = 3 and r = -2

Concept: Section Formula
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