Question

how has figure △def been transformed to form figure △abc? options: 180° rotation around (0,0), reflection across y - axis, translation 2 units down, 90° rotation around (0,0)

Explanation:

To determine the transformation from \(\triangle DEF\) to \(\triangle ABC\), we analyze each option:

• Reflection across \(y\)-axis: A reflection over the \(y\)-axis would map \((x,y)\) to \((-x,y)\), but the orientation and position don't match this.
• Translation 2 units down: This would shift the triangle vertically, but the triangles are not just shifted vertically; their orientation and position relative to the origin suggest a rotation.
• \(90^\circ\) rotation around \((0,0)\): A \(90^\circ\) rotation (either clockwise or counter - clockwise) would change the shape's orientation in a way that doesn't match the given triangles. For a counter - clockwise \(90^\circ\) rotation, \((x,y)\) maps to \((-y,x)\), and for clockwise, \((x,y)\) maps to \((y,-x)\), which is not the case here.
• \(180^\circ\) rotation around \((0,0)\): A \(180^\circ\) rotation around the origin maps a point \((x,y)\) to \((-x,-y)\). Visually, the triangle \(\triangle DEF\) when rotated \(180^\circ\) around the origin \((0,0)\) would have its vertices transformed in a way that aligns with \(\triangle ABC\). The orientation and position of the triangles suggest that a \(180^\circ\) rotation about the origin is the transformation.

Answer:

\(180^\circ\) rotation around \((0,0)\)