Question

after a rotation of 90° about the origin, the coordinates of the vertices of the image of a triangle are a(6, 3), b(-2, 1), and c(1, 7). what are the coordinates of the vertices of the pre - image?

Explanation:

Step1: Recall the rotation rule

When a point \((x,y)\) is rotated \(90^{\circ}\) clockwise about the origin, the transformation rule is \((x,y)\to(y, -x)\). To find the pre - image (reverse the rotation), if the image is \((x',y')\), then the pre - image \((x,y)\) is given by the rule \((x',y')\to(-y',x')\).

Step2: Find the pre - image of \(A'(6,3)\)

For \(A'(x' = 6,y'=3)\), using the rule \((x,y)=(-y',x')\).
\(x=-3\), \(y = 6\). So the pre - image of \(A'\) is \(A(-3,6)\).

Step3: Find the pre - image of \(B'(-2,1)\)

For \(B'(x'=-2,y' = 1)\), using the rule \((x,y)=(-y',x')\).
\(x=-1\), \(y=-2\). So the pre - image of \(B'\) is \(B(-1,-2)\).

Step4: Find the pre - image of \(C'(1,7)\)

For \(C'(x'=1,y' = 7)\), using the rule \((x,y)=(-y',x')\).
\(x=-7\), \(y = 1\). So the pre - image of \(C'\) is \(C(-7,1)\).

Answer:

\(A(-3,6)\), \(B(-1,-2)\), \(C(-7,1)\)