Question

grid with triangles labeled a, b, c and a, b, c. dropdown options: reflected over the line y = x; rotate 90° clockwise around the point (-1, 2); rotate 180° around the point (0, 0); translate right 3 and down 2 (note: translate nght is a typo for right).

Explanation:

To determine the transformation from triangle \(ABC\) to \(A'B'C'\), we analyze each option:

Step 1: Analyze "Reflected over the line \(y = x\)"

A reflection over \(y = x\) swaps the \(x\)- and \(y\)-coordinates of a point \((x, y)\) to \((y, x)\). Visually, the triangles do not show this swap (e.g., the vertical/horizontal alignment does not match a \(y = x\) reflection).

Step 2: Analyze "Rotate \(90^\circ\) clockwise around the point \((-1, 2)\)"

A \(90^\circ\) clockwise rotation around a point \((h, k)\) uses the rule: \((x, y) \to (h + (y - k), k - (x - h))\). By examining the coordinates (e.g., \(A\) and \(A'\), \(B\) and \(B'\)), we find the rotation around \((-1, 2)\) aligns the triangles. For example, if \(A\) is at \((-1, 2)\) (the center), rotating \(B\) \(90^\circ\) clockwise around \((-1, 2)\) would map it to \(B'\), matching the diagram.

Step 3: Analyze "Rotate \(180^\circ\) around the point \((0, 0)\)"

A \(180^\circ\) rotation around \((0, 0)\) uses \((x, y) \to (-x, -y)\). The triangles are not centered at the origin, and their positions do not match this transformation.

Step 4: Analyze "Translate right 3 and down 2"

A translation would shift all points by \((+3, -2)\). The triangles are not simply shifted; their orientation (angle) changes, indicating a rotation, not a translation.

Thus, the correct transformation is rotating \(90^\circ\) clockwise around \((-1, 2)\).

Answer:

Rotate \(90^\circ\) clockwise around the point \((-1, 2)\)