Question

the coordinates of the end points of a linear segment ab are a(6,10) and b(8,4). the line segment ab is translated 3 units to the left and 2 units up and then reflected over the line y = -x to obtain line segment ab. what are the coordinates of the end points of line segment ab? a. a(12,3) and b(6,5) b. a(3,12) and b(5,6) c. a(3, - 12) and b(5, - 6) d. a(-3,12) and b(-5,6) e. a(-12,-3) and b(-6,-5)

Explanation:

Step1: Perform the translation

For point \(A(6,10)\), translating 3 units to the left and 2 units up:
The \(x\) - coordinate of \(A\) becomes \(6 - 3=3\), and the \(y\) - coordinate becomes \(10 + 2 = 12\), so the new - point \(A_1\) is \((3,12)\).
For point \(B(8,4)\), translating 3 units to the left and 2 units up:
The \(x\) - coordinate of \(B\) becomes \(8 - 3 = 5\), and the \(y\) - coordinate becomes \(4+2 = 6\), so the new - point \(B_1\) is \((5,6)\).

Step2: Perform the reflection over \(y=-x\)

The rule for reflecting a point \((x,y)\) over the line \(y =-x\) is \((x,y)\to(-y,-x)\).
For point \(A_1(3,12)\), after reflection over \(y=-x\), the new point \(A'\) has coordinates \((-12,-3)\).
For point \(B_1(5,6)\), after reflection over \(y=-x\), the new point \(B'\) has coordinates \((-6,-5)\).

Answer:

E. \(A'(-12,-3)\) and \(B'(-6,-5)\)