Question

the point ( a(-6,-5) ) is reflected over the point ( (-5,1) ) 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 \(x_1=-6,y_1 = - 5,x_m=-5,y_m = 1\).

Step2: Solve for \(x_2\)

From \(x_m=\frac{x_1 + x_2}{2}\), substitute the values: \(-5=\frac{-6 + x_2}{2}\). Multiply both sides by \(2\): \(-10=-6 + x_2\). Then \(x_2=-10 + 6=-4\).

Step3: Solve for \(y_2\)

From \(y_m=\frac{y_1 + y_2}{2}\), substitute the values: \(1=\frac{-5 + y_2}{2}\). Multiply both sides by \(2\): \(2=-5 + y_2\). Then \(y_2=2 + 5 = 7\).

Answer:

$(-4,7)$