I have a central point and coordinates of a line and I need to do a central symmetry to get new coordinates. How can I do the central symmetry. Central point is (2.4, 2.8) Line coordinates are A(2,3)B(0,4)C(0,2)
Central symmetry using Central point and coordinates of a line
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geometry
trigonometry
2 Answers
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You want to rotate the points, given by $x+yi$ in the complex plane $180$ degrees about the center point. Note $e^{\pi i}=-1$. So all we have to do, to rotate about the origin, is multiply $x+yi$ by $-1$. This should make sense because if we multiply by $-1$ twice, that is equivalent to multiplying by $1$. A $360$ degree rotation, as we end up in the same place.
From,
$$(x+yi)(-1)=-x-yi$$
We have,
$$(x,y) \mapsto (-x,-y)$$
Before we apply the formula we have to translate all points so that $(2.4,2.8)$ becomes the origin. We do this by subtracting $2.4$ and $2.8$ from the $x$ and $y$ coordinates respectively. Then we apply the formula, and then reverse our translation by adding $2.4$ and $2.8$ to the $x$ and $y$ coordinates respectively.
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Hint: let the central point be $K(2.4,2.8)$, then any pair of symmetric points $P(x,y)$ and $P'(x',y')$ will have $K$ as the midpoint of the segment between them, so $\frac{1}{2}(x+x')=2.4\,$, $\frac{1}{2}(y+y')=2.8$.