If two vertices of a triangle are \(A(3, 1, 4)\) and \(B(-4, 5, -3)\) and the centroid of the triangle is \(G(-1, 2, 1)\), then find the coordinates of the third vertex \(C\) of the triangle.

Q.17 If two vertices of a triangle are \(A(3, 1, 4)\) and \(B(-4, 5, -3)\) and the centroid of the triangle is \(G(-1, 2, 1)\), then find the coordinates of the third vertex \(C\) of the triangle.

Doubtly Asked 28 Oct 2026 Board exam · March 2026 hsc 12th Save WhatsApp Report

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Answer: \(C(-2, 0, 2)\)

Let \(C = (x, y, z)\). The centroid is \(G = \left(\dfrac{3 - 4 + x}{3}, \dfrac{1 + 5 + y}{3}, \dfrac{4 - 3 + z}{3}\right) = (-1, 2, 1)\).

\(\dfrac{x - 1}{3} = -1 \Rightarrow x = -2\); \(\dfrac{y + 6}{3} = 2 \Rightarrow y = 0\); \(\dfrac{z + 1}{3} = 1 \Rightarrow z = 2\).

Hence the third vertex is \(C(-2, 0, 2)\).

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report

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If two vertices of a triangle are \(A(3, 1, 4)\) and \(B(-4, 5, -3)\) and the centroid of the triangle is \(G(-1, 2, 1)\), then find the coordinates of the third vertex \(C\) of the triangle.

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