Question

the point ( a(4,-8) ) is reflected over the point ( (1,-1) ) and its image is point ( b ). what are the coordinates of point ( b )?

Explanation:

Step1: Use the mid - point formula

If a point \(A(x_1,y_1)\) is reflected over a point \(M(x_m,y_m)\) to get a point \(B(x_2,y_2)\), then the mid - point formula \(x_m=\frac{x_1 + x_2}{2}\) and \(y_m=\frac{y_1 + y_2}{2}\) holds. Here \(x_1 = 4,y_1=-8,x_m = 1,y_m=-1\).
For the \(x\) - coordinate:
We have \(1=\frac{4 + x_2}{2}\).
Multiply both sides by \(2\): \(2=4 + x_2\).
Subtract \(4\) from both sides: \(x_2=2 - 4=-2\).

Step2: For the \(y\) - coordinate

We have \(-1=\frac{-8 + y_2}{2}\).
Multiply both sides by \(2\): \(-2=-8 + y_2\).
Add \(8\) to both sides: \(y_2=-2 + 8 = 6\).

Answer:

\((-2,6)\)