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The three vertices of a rhombus PQRS are P(2, 3), Q(6, 5) and R(-2, 1). Find the coordinates of the fourth vertex S and coordinates of the point where both the diagonals PR and QS intersect.
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Let the coordinates of fourth vertex S be $(x,y)$
Coordinates of mid-point of PR = Coordinates of mid-point of QS
$(\frac{x+6}{2}, \frac{y+5}{2}) = (\frac{2-2}{2}, \frac{3+1}{2})$ (1 Mark)
x = -6 (1/2 Mark)
y = -7 (1/2 Mark)
$\therefore$ coordinates of S = (-6, -7)
mid- point of diagonal PR = (0, -1) (1 Mark)
Coordinates of mid-point of PR = Coordinates of mid-point of QS
$(\frac{x+6}{2}, \frac{y+5}{2}) = (\frac{2-2}{2}, \frac{3+1}{2})$ (1 Mark)
x = -6 (1/2 Mark)
y = -7 (1/2 Mark)
$\therefore$ coordinates of S = (-6, -7)
mid- point of diagonal PR = (0, -1) (1 Mark)