Question

The orthocenter of the triangle having vertices as (2, 3), (2,5),(4,3) is​

Answer 1

we have to find orthocenter of the triangle having vertices as (2, 3) , (2, 5) and (4, 3).

Solution : orthocentre is the point of intersection of altitudes of triangle.

so if we find equation of two altitudes, we can get orthocentre by solving it.

let A(2, 3) , B(2, 5) and C(4, 3)

slope of AB = (5 - 3)/(2 - 2) = 1/0

so slope of altitude on AB = -1/slope of AB = 0

equation of altitude on AB which is passing through C(4,3) is given by,

(y - 3) = 0(x - 4)

⇒y = 3 .......(1)

again, slope of BC = (3 - 5)/(4 - 2) = -2/2 = -1

so slope of altitude on BC = -1/slope of BC = 1

now equation of altitude on BC which is passing through A is given by,

(y - 3) = 1(x - 2)

⇒x - y + 1 = 0......(2)

from equations (1) and (2) we get,

y = 3 and x = 2

therefore (2,3) is the orthocentre of given triangle.