Question

a point is rotated 270° about the origin. the image of the point is (-11,7). what are the coordinates of the preimage?
choose the correct answer below.
a. (-7,-11)
b. (7,11)
c. (11,7)
d. (7,-11)

Explanation:

Step1: Recall the rotation rule

When a point \((x,y)\) is rotated \(270^{\circ}\) counter - clockwise about the origin, the transformation rule is \((x,y)\to(y, - x)\). Let the pre - image be \((a,b)\). After a \(270^{\circ}\) counter - clockwise rotation about the origin, the image is \((b,-a)\).
We know the image is \((-11,7)\). So, if \(b=-11\) and \(-a = 7\), then \(a=-7\).

Step2: Find the pre - image

If we consider the rotation in the reverse direction (a \(90^{\circ}\) clockwise rotation is equivalent to a \(270^{\circ}\) counter - clockwise rotation). The rule for a \(90^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y,-x)\). Let the pre - image be \((x,y)\) and the image be \((X,Y)\). We have \(X = y\) and \(Y=-x\). Given \(X=-11\) and \(Y = 7\), then \(y=-11\) and \(x=-7\).

Answer:

A. \((-7,-11)\)