Question

which one of the following images shows the pre-image and image of a triangle which has been rotated 180° about the origin? a image, b image, c image, d image

Explanation:

To determine which image shows a triangle rotated \(180^\circ\) about the origin, we use the property of a \(180^\circ\) rotation: for a point \((x, y)\), the image after a \(180^\circ\) rotation about the origin is \((-x, -y)\). This means the pre - image and image should be centrally symmetric about the origin, with corresponding vertices being opposite in both the \(x\) - and \(y\) - coordinates.

Analyzing Option A

In option A, the positions of the triangles do not show the central symmetry about the origin. The relative positions of the vertices do not satisfy the \((x,y)\to(-x, -y)\) transformation.

Analyzing Option B

In option B, the triangles are not centrally symmetric about the origin. The coordinates of the corresponding vertices do not follow the rule of \(180^\circ\) rotation.

Analyzing Option C

For a triangle rotated \(180^\circ\) about the origin, if we consider the vertices of the pre - image and the image, the pre - image (pink) and the image (blue) in option C have vertices that are related by the transformation \((x,y)\to(-x, -y)\). The triangle is flipped both horizontally and vertically with respect to the origin, which is consistent with a \(180^\circ\) rotation.

Analyzing Option D

In option D, the triangles do not show the correct central symmetry about the origin. The relative positions of the vertices do not match the \(180^\circ\) rotation transformation.

Answer:

C