Question

figure h is the result of a transformation on figure g. which transformation would accomplish this?
answer
a translation 4 units to the left and 2 units down
a reflection over the x - axis
a rotation 180° counterclockwise about the origin
a rotation 90° clockwise about the origin

Explanation:

Step1: Analyze translation

Translation moves the figure without rotation or reflection. If we consider translation 4 units left and 2 units down, the orientation of the figure (like the slope of the sides) would remain the same. But Figure H has a different orientation compared to Figure G if we just consider a simple translation.

Step2: Analyze reflection over \(x -\)axis

Reflection over \(x -\)axis changes the \(y -\)coordinate (\((x,y)\to(x, - y)\)). If we reflect Figure G over \(x -\)axis, the relative positions of the vertices with respect to the \(x -\)axis would be inverted. But Figure H is not a simple reflection over \(x -\)axis of Figure G.

Step3: Analyze rotation \(180^{\circ}\) counter - clockwise about the origin

The rule for a \(180^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-x,-y)\). Let's assume a vertex of Figure G is \((x,y)\). After \(180^{\circ}\) rotation, it becomes \((-x,-y)\). If we check the general position and orientation (for example, if one vertex of Figure G is \((2,1)\), after \(180^{\circ}\) rotation about the origin it is \((- 2,-1)\), if another vertex is \((0,4)\) it becomes \((0, - 4)\)). The orientation (the way the figure is "facing") is reversed in a way that matches Figure H.

Step4: Analyze rotation \(90^{\circ}\) clockwise about the origin

The rule for a \(90^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y,-x)\). This would change the orientation in a different way than what is observed from Figure G to Figure H. For example, if a point \((x,y)=(2,1)\) after \(90^{\circ}\) clockwise rotation is \((1,-2)\), which does not match the general transformation from Figure G to Figure H.

Answer:

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