Question

If the vertices of a triangle are $(a, 1)$, $(b, 3)$, and $(4, c)$, then the centroid of the triangle will lie on the x-axis if:

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Answer

The centroid of a triangle is found by averaging the coordinates of its vertices. In this case, the vertices are given as $(a, 1)$, $(b, 3)$, and $(4, c)$. To find the centroid, you calculate the average of the $x$-coordinates and the average of the $y$-coordinates separately.

The formula for the centroid $(X, Y)$ is: $$ X = \frac{a + b + 4}{3}, \quad Y = \frac{1 + 3 + c}{3} $$

Given that the centroid should lie on the x-axis, the $y$-coordinate of the centroid, $Y$, must equal zero. This condition gives us: $$ \frac{1 + 3 + c}{3} = 0 $$

Solving this equation for $c$: $$ 1 + 3 + c = 0 \ c = -4 $$

Therefore, the value of $c$ that ensures the centroid lies on the x-axis is $c = -4$.


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