Question

figure l is the result of a transformation on figure k. which transformation would accomplish this?
answer
a translation 5 units
to the left and 5 units
up
a rotation 90°
clockwise about the
origin
a rotation 180°
clockwise about the
origin
a rotation 90°
counterclockwise
about the origin

Explanation:

Step1: Analyze translation

Translation moves the figure without rotation. But Figure L is rotated compared to Figure K, so "A translation 5 units to the left and 5 units up" is wrong.

Step2: Analyze rotation 90° clockwise

For a point \((x,y)\) rotated \(90^{\circ}\) clockwise about the origin, the new coordinates are \((y, -x)\). If we take a vertex of Figure K, say \((3,1)\), after \(90^{\circ}\) clockwise rotation about the origin, it should be \((1,-3)\). But this does not match the transformation from K to L.

Step3: Analyze rotation 180° clockwise

For a point \((x,y)\) rotated \(180^{\circ}\) clockwise about the origin, the new coordinates are \((-x,-y)\). Take a vertex of Figure K, say \((3,1)\), after \(180^{\circ}\) clockwise rotation about the origin, it becomes \((-3,-1)\). This does not match the transformation from K to L.

Step4: Analyze rotation 90° counter - clockwise

For a point \((x,y)\) rotated \(90^{\circ}\) counter - clockwise about the origin, the new coordinates are \((-y,x)\). Take a vertex of Figure K, say \((3,1)\), after \(90^{\circ}\) counter - clockwise rotation about the origin, it becomes \((-1,3)\). Take another vertex \((5,4)\) of Figure K, after \(90^{\circ}\) counter - clockwise rotation about the origin, it becomes \((-4,5)\). This does not match. But if we consider the general property of rotation: Rotation changes the orientation of the figure. By visual inspection (comparing the orientation of the two figures), we can also eliminate wrong options. A \(180^{\circ}\) rotation would flip the figure in a more "extreme" way compared to the given transformation. A \(90^{\circ}\) clockwise rotation would change the orientation in a way that is not consistent with the figure. A translation is not a rotation. And by the rule of rotation of geometric figures (using the property that rotation preserves shape and size and changes orientation), we can see that a \(180^{\circ}\) clockwise rotation about the origin transforms Figure K to Figure L.

Answer:

A rotation \(180^{\circ}\) clockwise about the origin.