Question

q) which transformation will not carry the rectangle onto itself?
image of a coordinate grid with a red rectangle, and four multiple - choice options: rotation 180 degrees about the origin; rotation 180 degrees about (-1, 0); reflection over the x - axis; reflection over x = -1

Explanation:

Step1: Analyze Rotation 180° about Origin

A 180° rotation about the origin \((x,y)\to(-x,-y)\). For a rectangle, this swaps opposite corners, and if symmetric about origin (or structure allows), it maps to itself. The given rectangle, when rotated 180° about origin, will have its vertices transformed such that it aligns with itself (since it's symmetric in a way that 180° rotation preserves its shape and position relative to origin? Wait, actually, let's check coordinates. Suppose the rectangle has vertices (let's assume approximate coordinates: say left side at \(x = -2\), right at \(x = 0\), top at \(y = 2\), bottom at \(y = -2\)? Wait, no, looking at the grid, the rectangle is centered around \(x=-1\) maybe? Wait, maybe better to check each transformation:

Step2: Analyze Rotation 180° about \((-1,0)\)

A 180° rotation about a point \((h,k)\) transforms \((x,y)\to(2h - x, 2k - y)\). For \((h,k)=(-1,0)\), this is \((-2 - x, -y)\). If the rectangle is symmetric about \(x=-1\) (vertical line \(x=-1\)), then rotating 180° about \((-1,0)\) (which is on the vertical line of symmetry) will map the rectangle to itself.

Step3: Analyze Reflection over x - axis

Reflection over x - axis: \((x,y)\to(x,-y)\). The original rectangle has a top part and bottom part. If we reflect over x - axis, the top half (above x - axis) will go below, and bottom half (below x - axis) will go above. But looking at the rectangle, it's not symmetric over x - axis. Wait, the rectangle in the graph: let's see, the top is above x - axis, bottom below? Wait, no, maybe the rectangle is from, say, \(y = 1\) to \(y=-3\) (approx) and \(x=-2\) to \(x = 0\). Reflecting over x - axis would flip the y - coordinates. So a point \((-2,1)\) would go to \((-2,-1)\), and \((-2,-3)\) would go to \((-2,3)\). But the original rectangle doesn't have a point at \((-2,3)\), so this reflection would NOT map the rectangle to itself. Wait, but let's check other options.

Step4: Analyze Reflection over \(x=-1\)

Reflection over vertical line \(x = a\) is \((x,y)\to(2a - x, y)\). For \(a=-1\), it's \((-2 - x, y)\). This is a vertical reflection over \(x=-1\), which is the vertical line of symmetry for the rectangle (since the rectangle is centered at \(x=-1\), left and right sides are equidistant from \(x=-1\)). So this reflection will map the rectangle to itself.

Now, going back: Rotation 180° about origin: Let's take a vertex, say \((-2,2)\) (hypothetical), rotating 180° about origin gives \((2,-2)\), which is not on the original rectangle. Wait, maybe my coordinate assumption is wrong. Wait, the key is: reflection over x - axis. The rectangle has a "top" and "bottom" that are not mirror images over x - axis. So reflection over x - axis will not carry the rectangle onto itself. The other transformations (rotation 180° about origin? Wait, maybe I made a mistake earlier. Wait, no, let's re - evaluate.

Wait, the correct answer is "Reflection over the x - axis" because:

- Rotation 180° about origin: For a rectangle that is symmetric in a way that 180° rotation (swapping x and -x, y and -y) works? Wait, maybe the rectangle is centered at the origin? No, the vertical line of symmetry is \(x=-1\). Wait, maybe the initial analysis was wrong. Let's think again:

The rectangle is symmetric about the vertical line \(x=-1\) (so reflection over \(x=-1\) works). Rotation 180° about \((-1,0)\) (a point on \(x=-1\)) will work because 180° rotation about a point on the axis of symmetry will map the figure to itself. Rotation 180° about origin: if the rectangle is not centered at origin, but centered at \(x=-1\), then rotation about origin will move it, but maybe in this case, the rectangle's vertices, when rotated 180° about origin, still form the same rectangle? Wait, no, probably the reflection over x - axis is the one that doesn't work. Because the rectangle has a top and bottom that are not congruent in terms of their position relative to x - axis (maybe the top is shorter or longer? Wait, the graph shows a rectangle that is symmetric vertically (over \(x=-1\)) but not horizontally (over x - axis). So reflecting over x - axis will flip it vertically, and the resulting figure will not match the original.

Answer:

Reflection over the x - axis