# If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and G(r,-43,13) is its centroid then find the values of p, q and r - Mathematics and Statistics

Sum

If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and "G"("r", -4/3, 1/3) is its centroid then find the values of p, q and r

#### Solution

Let bar"a", bar"b", bar"c" be the position vectors of points A, B, C respectively of ∆ABC and bar"g" be the position vector of its centroid G.

∴ bar"a" = 5hat"i" + hat"j" + "p"hat"k",

bar"b" = hat"i" + "q"hat"j" + "p"hat"k",

bar"c" = hat"i" - 2hat"j" + 3hat"k"

and

bar"g" = "r"hat"i" - 4/3hat"j" + 1/3hat"k"

∴ By using centroid formula,

bar"g" = (bar"a" + bar"b" + bar"c")/3

∴ 3bar"g" = bar"a" + bar"b" + bar"c"

∴ 3("r"hat"i" - 4/3hat"j" + 1/3hat"k") = (5hat"i" + hat"j" + "p"hat"k") + (hat"i" + "q"hat"j" + "p"hat"k") + (hat"i" - 2hat"j + 3hat"k")

∴ 3"r"hat"i" - 4hat"j" + hat"k" = 7hat"i" + ("q" - 1)hat"j" + (2"p" + 3)hat"k"

∴ By equality of vectors, we get

3r = 7, – 4 = q – 1 and 1 = 2p + 3

∴ r = 7/3, q = – 3 and p = – 1

Concept: Section Formula
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