Question

the point ( a(3,-6) ) is reflected over the point ( (2,-1) ) and its image is point ( b ). what are the coordinates of point ( b )?

Explanation:

Step1: Recall 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 \(x_1 = 3,y_1=-6,x_m = 2,y_m=-1\).

Step2: Solve for \(x_2\)

Using \(x_m=\frac{x_1 + x_2}{2}\), substitute the values: \(2=\frac{3 + x_2}{2}\). Multiply both sides by 2: \(4=3 + x_2\). Then \(x_2=4 - 3=1\).

Step3: Solve for \(y_2\)

Using \(y_m=\frac{y_1 + y_2}{2}\), substitute the values: \(-1=\frac{-6 + y_2}{2}\). Multiply both sides by 2: \(-2=-6 + y_2\). Then \(y_2=-2 + 6 = 4\).

Answer:

$(1,4)$