Home
Class 12
MATHS
Prove that the circumcenter, orthocentre...

Prove that the circumcenter, orthocentre, incenter, and centroid of the triangle formed by the points `A(-1,11),B(-9,-8),` and `C(15 ,-2)` are collinear, without actually finding any of them.

Text Solution

Verified by Experts

The given points are `A(-1,11),B(-9,-8)`, and `C(15,-2)`.
`AB=sqrt(64+361)=sqrt(425)`
`BC=sqrt(576+36)=sqrt(612)`
`AC=sqrt(256+169)=sqrt(425)`
Thus, `AB=ACneBC`.
Therefore, triangle is isosceles, and in an isosceles triangle, all centres are collinear.
Promotional Banner

Topper's Solved these Questions

  • COORDINATE SYSYEM

    CENGAGE ENGLISH|Exercise Concept applications 1.1|6 Videos

  • COORDINATE SYSYEM

    CENGAGE ENGLISH|Exercise Concept applications 1.2|8 Videos

  • COORDINATE SYSYEM

    CENGAGE ENGLISH|Exercise Illustration1.67|1 Videos

  • COORDINATE SYSTEM

    CENGAGE ENGLISH|Exercise Multiple Correct Answers Type|2 Videos

  • CROSS PRODUCTS

    CENGAGE ENGLISH|Exercise DPP 2.2|13 Videos

Similar Questions

Explore conceptually related problems

Find the centroid of the triangle formed by the points (2,7) (3,-1)(-5,6).

Prove that the points A(9,-1,4),B(-1,-3,2) and C(4,-2,3) are collinear.

Show that the points A(6,4) , B(9,7) and C(11,9) are collinear.

The points A(-3, 2), B(2, -1) and C(a, 4) are collinear. Find a.

Find the centroid of the triangle whose vertices are A(-1,0), B(5, -2) and C(8, 2).

Prove that the points (2,\ -2),\ (-3,\ 8) and (-1,\ 4) are collinear.

Show that the point A(2,-3,-4), B(1,2,3), C(3,-8,-11) are collinear.

Prove that the centroid of any triangle is the same as the centroid of the triangle formed by joining the middle points of its sides

Using vectors, prove that the points (2,-1,3) ,(3,-5,1) and (-1,11,9) are collinear

Find the orthocentre of the triangle ABC whose angular points are A(1,2),B(2,3) and C(4,3)