Question

If the centroid of the triangle formed by the points \((3,-5),(-7,4)\) and \((10,-k )\) is at the point \((k, –1)\), then the value of k is

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

To find the value of \( k \) for the centroid of the triangle formed by the points \((3,-5)\), \((-7,4)\), and \((10,-k)\), which is given to be at \((k, -1)\), we use the formula for the centroid \((x_c, y_c)\) of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\):
\((x_c, y_c) = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)\).
Substituting the given points and the centroid coordinates:
\((x_c, y_c) = \left(\frac{3 + (-7) + 10}{3}, \frac{-5 + 4 + (-k)}{3}\right) = (k, -1)\).
Solving for the x-coordinate of the centroid:
\(\frac{3 - 7 + 10}{3} = k\)
\(\frac{6}{3} = k\)
\(k = 2\)
Solving for the y-coordinate of the centroid:
\(\frac{-5 + 4 - k}{3} = -1\)
\(-1 - k = -3\)
\(k = 2\)
Therefore, the value of \(k\) is 2.

Answer 1

The Correct Option is B

Given: The vertices of the triangle are \((3,-5)\), \((-7,4)\), and \((10,-k)\). 

The centroid \( G(x_g, y_g) \) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:

\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

Step 1: Apply centroid formula

\[ \left( \frac{3 + (-7) + 10}{3}, \frac{-5 + 4 + (-k)}{3} \right) = (k, -1) \]

Step 2: Solve for \( k \)

\[ \frac{3 - 7 + 10}{3} = k \] \[ \frac{6}{3} = k \] \[ k = 2 \]

Final Answer: 2