Question

multiple choice 6 points
triangle abc is congruent to triangle abc by a rigid transformation. choose below the rigid transformation that proves its congruence.
translation 2 units left followed by a 90 degree counterclockwise rotation around the origin
rotation 180 degree counterclockwise around the origin
rotation 90 degrees counterclockwise around the origin
reflection over the x - axis then a translation 2 units left

Explanation:

Step1: Recall transformation rules

Rigid - transformations include translations, rotations, and reflections. We need to analyze each option on the given triangles.

Step2: Analyze option 1

Translation 2 units left: \((x,y)\to(x - 2,y)\). Then 90 - degree counter - clockwise rotation around the origin \((x,y)\to(-y,x)\). This does not map \(\triangle ABC\) to \(\triangle A'B'C'\).

Step3: Analyze option 2

For a 180 - degree counter - clockwise rotation around the origin, the rule is \((x,y)\to(-x,-y)\). If we take the vertices of \(\triangle ABC\) and apply this rule, we can see that it maps \(\triangle ABC\) to \(\triangle A'B'C'\). For example, if \(A=( - 1,1)\), after 180 - degree rotation, \(A'=(1, - 1)\).

Step4: Analyze option 3

A 90 - degree counter - clockwise rotation around the origin has the rule \((x,y)\to(-y,x)\). This does not map \(\triangle ABC\) to \(\triangle A'B'C'\).

Step5: Analyze option 4

Reflection over the x - axis has the rule \((x,y)\to(x,-y)\), then translation 2 units left \((x,y)\to(x - 2,y)\). This does not map \(\triangle ABC\) to \(\triangle A'B'C'\).

Answer:

Rotation 180 degree counterclockwise around the origin