Question

# If in triangle , circumcenter and orthocenter then the co-ordinates of mid-point of side opposite to is :

- (1, -11/3)
- (1,5)
- (1, -3)
- (1,6)

Hint:

### Centroid, orthocenter, circumcenter are collinear. The centroid divides the median in 2:1 ratio.

## The correct answer is: (1, -11/3)

### Given That:

If in triangle , circumcenter and orthocenter then the co-ordinates of mid-point of side opposite to is :

>>>The orthocenter, centroid and circumcenter of any triangle are collinear. And the centroid divides the distance from orthocenter to circumcenter in the ratio $2:1$.

>>Let the centroid be $G(x,y)$ , its coordinates can be found using the section formula. Then,
>>> Also, the centroid $(G)$ divides the medians $(AD)$ in the ratio $2:1. Then:$
>>>Let the coordinates of $D$ be $(h,k)$
$h=1$ and $6k+30=8$
and k =
$∴D(h,k)=(1, )$

>>> The orthocenter, centroid and circumcenter of any triangle are collinear. And the centroid divides the distance from orthocenter to circumcenter in the ratio $2:1$.

>>> Also, the centroid $(G)$ divides the medians $(AD)$ in the ratio $2:1.$

>>> ∴D(h, k)=(1,)

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Reason (R): The above three lines concur at (1,1)

Both Assertion and Reason are correct and the Reason is the correct explanation of Assertion.

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Reason (R): The above three lines concur at (1,1)

Both Assertion and Reason are correct and the Reason is the correct explanation of Assertion.

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