Question

The co-ordinates of the centroid of a triangle are (2, −5). If two of its vertices are (−6, 5) and (11, 8), find the coordinates of the third vertex.

Answer 1

The formula used is the centroid (intersection of the medians) of triangle ABC:

G = `((x_1 + x_2 + x_3) / 3, (y_1 + y_2 + y_3) / 3),`

Because the centroid divides each median in the ratio 2 : 1,

To find the missing vertex (x₃,y₃) when G and the other two vertices (x₁,y₁), (x₂,y₂) are known,

x₃ = 3Gx − x₁ − x₂,

y₃ = 3Gy − y1 − y₂

Applying this to G = (2, −5), A = (−6, 5), B = (11, 8):

⇒ x₃ = 3 × 2 − (−6) − 11

= 6 + 6 − 11

∴ x₃ = 1

⇒ y₃ = 3 × (−5) − 5 − 8

= −15 − 13

∴ y₃ = −28

Hence, the third vertex is (1, −28).