If the two vertices of a triangle are (5, 4), and (-2, 4) and the centroid is (5, 6), then the third vertex is:

(12, 10)

The correct answer is Option (4) → (12, 10)

Let the third vertex be (x, y).

The centroid $(G_x, G_y)$ of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is given by:

$\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$

Given:

• First vertex = (5, 4)
• Second vertex = (-2, 4)
• Centroid = (5, 6)

So,

$\frac{5 + (-2) + x}{3} = 5$

$\frac{4 + 4 + y}{3} = 6$

Solve for x:

$\frac{3 + x}{3} = 5 \Rightarrow 3 + x = 15 \Rightarrow x = 12$

Solve for y:

$\frac{8 + y}{3} = 6 \Rightarrow 8 + y = 18 \Rightarrow y = 10$

So, the third vertex is: $(12, 10)$