Maharashtra State BoardHSC Science (General) 11th
Advertisement Remove all ads

Find the coordinates of the orthocenter of the triangle whose vertices are A(2, −2), B(1, 1), and C(−1, 0). - Mathematics and Statistics

Sum

Find the coordinates of the orthocenter of the triangle whose vertices are A(2, −2), B(1, 1), and C(−1, 0).

Advertisement Remove all ads

Solution


Let AM and BN be the altitudes of the ΔABC.

Now, slope of BC = `(0 - 1)/(-1 - 1) = 1/2`

Altitude AM a perpendicular to side BC.

∴ slope of altitude AM = – 2 and it is passing through A(2, –2).

∴ equation of the altitude AM is y – (– 2) = – 2 (x – 2)

∴ y + 2 = – 2x + 4

∴ 2x + y = 2     ...(1)

Slope of side AC = `(0 - (- 2))/(-1 - 2) = -2/3` 

Altitude BN is perpendicular to side AC.

∴ slope of altitude BN = `3/2` and it is passing through B (1, 1).

∴ equation of the altitude BN is y – 1 = `3/2(x - 1)`

∴ 2y – 2 = 3x – 3

∴ 3x – 2y = 1   ...(2)

The orthocentre H is the point of intersection of the altitudes AM and BN. Hence, we solve equations (1) and (2).

Multiply equation (1) by 2, we get,

4x + 2y = 4    ...(3)

Adding (2) and (3), we get,

7x = 5

∴  x = `5/7`

∴  from (1), `2(5/2) + y` = 2

∴ y = `2 - 10/7`

= `4/7`

Hence, coordinates of orthocentre H are `(5/7, 4/7)`.

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×