Find the coordina

Given: coordinates of the orthocenter of the triangle whose vertices are ( - 1, 3), (2, - 1) and (0, 0).

Assuming:

A (0, 0), B (−1, 3) and C (2, −1) be the vertices of the triangle ABC.

Let AD and BE be the altitudes.

To find:

Orthocenter of the triangle.

Explanation:

AD⊥BC and BE⊥AC

∴ The slope of AD × Slope of BC = −1

The slope of BE × Slope of AC = −1

Here, the slope of BC =

and slope of AC =

∴ slope of AD × ( - 4/3) = - 1 and slope of BE × ( - 1/2) = - 1

⇒ slope of AD and slope of BE = 2

The equation of the altitude AD passing through A (0, 0) and having slope is

y - 0 ( x - 0)

⇒ y x …..(1)

The equation of the altitude BE passing through B (−1, 3) and having slope 2 is

y - 3 = 2(x + 1)

⇒ 2x – y + 5 = 0 …….(2)

Solving (1) and (2):

x = − 4, y = − 3

Hence, the coordinates of the orthocentre is (−4, −3).