First slide

Cartesian plane

Question

If in triangle ABC, A = (1, 10), circumcenter = (-1/3,2/3), and orthocenter = (11/4 , 4/3), then the coordinates of the midpoint of the side opposite to A are

Moderate

A

(1, -11/3)
B

(1, 5)
C

(1, -3)
D

(1,6)

Solution

Circumcenter O=(-1/3, 2/3) and orthocenter H = (11/3,4/3).
Therefore, the coordinates of G are (1,8/9). Now, the point
A is (1, 10) as G is (1 ,8/9). Hence,
AD:DG = 3:1
$∴ D_{x} = \frac{3 - 1}{2} = 1, D_{y} = \frac{(8 / 3) - 10}{2} = - \frac{11}{3}$
Hence, the coordinates of the midpoint of BC are (1, -11/3).

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