Question

a student claims that rotating a shape 180° and reflecting it over the x - axis will result in the same final position. do you agree? explain your reasoning using an example or coordinate rule.

Explanation:

Rotation of a point \((x,y)\) by \(180^{\circ}\) gives \((-x,-y)\). Reflection of a point \((x,y)\) over the \(x -\)axis gives \((x,-y)\). These are different transformations. For example, take the point \((1,2)\). Rotating \(180^{\circ}\) gives \((-1,-2)\). Reflecting over \(x -\)axis gives \((1,-2)\).

Answer:

No, I do not agree. Rotating a point \((x,y)\) by \(180^{\circ}\) gives \((-x,-y)\) (using the coordinate rule \((x,y)\to(-x,-y)\) for \(180^{\circ}\) rotation). Reflecting a point \((x,y)\) over the \(x -\)axis gives \((x,-y)\) (using the coordinate rule \((x,y)\to(x, - y)\) for \(x -\)axis reflection). For example, if we have the point \((1,2)\):

• After \(180^{\circ}\) rotation: \((1,2)\to(-1,-2)\)
• After \(x -\)axis reflection: \((1,2)\to(1,-2)\)

Since \((-1,-2)
eq(1,-2)\), the two transformations do not result in the same final position.