Question

what are the coordinates of the vertices of the the image of rectangle jklm after a dilation centered at the origin with a scale factor of 3 followed by a rotation 270° counterclockwise about the origin?

Explanation:

First, we need to determine the original coordinates of the vertices of rectangle \(JKLM\). From the graph:

• \(J(-1, 1)\)
• \(K(0, 1)\)
• \(L(0, -1)\)
• \(M(-1, -1)\)

Step 1: Dilation with a scale factor of 3 centered at the origin

The rule for dilation centered at the origin with a scale factor \(k\) is \((x, y) \to (kx, ky)\).

• For \(J(-1, 1)\): \(J' = (3\times(-1), 3\times1) = (-3, 3)\)
• For \(K(0, 1)\): \(K' = (3\times0, 3\times1) = (0, 3)\)
• For \(L(0, -1)\): \(L' = (3\times0, 3\times(-1)) = (0, -3)\)
• For \(M(-1, -1)\): \(M' = (3\times(-1), 3\times(-1)) = (-3, -3)\)

Step 2: Rotation of \(270^\circ\) counterclockwise about the origin

The rule for a \(270^\circ\) counterclockwise rotation about the origin is \((x, y) \to (y, -x)\).

• For \(J'(-3, 3)\): \(J'' = (3, -(-3)) = (3, 3)\)
• For \(K'(0, 3)\): \(K'' = (3, -0) = (3, 0)\)
• For \(L'(0, -3)\): \(L'' = (-3, -0) = (-3, 0)\)
• For \(M'(-3, -3)\): \(M'' = (-3, -(-3)) = (-3, 3)\)

Answer:

The coordinates of the vertices of the image after dilation and rotation are:

• \(J''(3, 3)\)
• \(K''(3, 0)\)
• \(L''(-3, 0)\)
• \(M''(-3, 3)\)