Question

Find the orthocenter of the triangle whose vertices are (-2,-1),(6,-1)and (2,5)

Answer 1

orthocenter is obtained by the intersection of altitudes of a triangle

Answer 2

The orthocentre of the triangle is (2, 5/3).

We have to find the orthocentre of the triangle whose vertices are (-2, -1) , (6, -1) and (2, 5).

Concept : Orthocentre is the intersecting point of all altitudes of a triangle.

To find Orthocentre,

• first find equations of at least two altitudes.
• now find intersecting point of these two altitudes.

let A(-2, -1), B(6, -1) , C(2, 5)

∵ slope of line perpendicular to AB × slope of AB = -1

⇒slope of line perpendicular to AB × (-1 + 1)/(6 +2) = -1

⇒slope of line perpendicular to AB = 1/0

now equation of line perpendicular to AB drawn from C ;

(y - 5) = (1/0) (x - 2)

⇒x = 2 ...(1)

again, slope of line perpendicular to BC × slope of line BC = -1

⇒slope of line perpendicular to BC × (5 + 1)/(2 - 6) = -1

⇒slope of line perpendicular to BC × 6/-4 = -1

⇒slope of line perpendicular to BC = 2/3

now equation of line perpendicular to BC drawn from A ;

(y + 1) = 2/3(x + 2)

⇒3(y + 1) = 2(x + 2) ...(2)

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

x = 2 and y = 5/3

Therefore the orthocentre of the triangle is (2, 5/3).