the centroid and circumcentre of triangle is3,3 and 6,2respectively find orthocentre
Answers
Answer:
Kya Kya question puchte Ho yarr
Step-by-step explanation:
In any triangle, orthocentre, centroid and circumcentre are
collinear and centroid divides the line joining orthocentre and circumcenter in
the ratio 2 : 1.
Let the orthocentre be (x,y) (x,y)
Using the section formula, if a point
(x,y)(x,y) divides the line joining the points ({ x }_{ 1 },{ y }_{ 1 })(x
1
,y
1
) and
({ x }_{ 2 },{ y }_{ 2 })(x
2
,y
2
) in the ratio m:n m:n, then $$(x,y) = \left(
\dfrac { m{ x }_{ 2 } + n{ x }_{ 1 } }{ m + n } ,\dfrac { m{ y }_{ 2 } + n{
y }_{ 1 } }{ m + n } \right) $$
Substituting $$({ x }_{ 1 },{ y }_{
1 }) = (x,y) and and({x }_{ 2 },{ y }_{ 2 }) = (6,2) and and m = 2, n
= 1 in the section formula, we get the centroid inthesectionformula,wegetthecentroid = \left( \dfrac {
2(6) + 1(x) }{ 2 +1 } ,\dfrac { 2(2) + 1(y) }{ 2 + 1 } \right) =
=\left( \dfrac { x + 12 }{ 3 } ,\dfrac { y + 4 }{ 3} \right) $$
Given
centroid = (3,3) =(3,3)
=> \left( \dfrac { x + 12 }{ 3 } ,\dfrac { y + 4 }{ 3} \right) = (3,3) =>(
3
x+12
,
3
y+4
)=(3,3)
=> \dfrac { x + 12 }{ 3 } = 3 ; \dfrac { y + 4 }{ 3} = 3 =>
3
x+12
=3;
3
y+4
=3
x + 12 = 9 ; y + 4 = 9 x+12=9;y+4=9
x = -3 ; y = 5 x=−3;y=5
Hence, orthocentre = (-3,5) =(−3,5)