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

The centroid of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is:

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

Given:

• First vertex: $(5, 4)$
• Second vertex: $(-2, 4)$
• Centroid: $(5, 6)$
• Third vertex: $(x_3, y_3)$ to be found

Using the centroid formula for the $x$-coordinate:

$\dfrac{x_1 + x_2 + x_3}{3} = 5$

$\dfrac{5 + (-2) + x_3}{3} = 5$

$\dfrac{3 + x_3}{3} = 5$

$3 + x_3 = 15$

$x_3 = 12$

Using the centroid formula for the $y$-coordinate:

$\dfrac{y_1 + y_2 + y_3}{3} = 6$

$\dfrac{4 + 4 + y_3}{3} = 6$

$\dfrac{8 + y_3}{3} = 6$

$8 + y_3 = 18$

$y_3 = 10$

Therefore, the third vertex is $(12, 10)$.