Question

find the image of the quadrilateral to the right in a 180° counterclockwise rotation about d. choose the correct figure of the rotation below. a. b. c.

Explanation:

Step1: Understand the rotation rule

A \(180^{\circ}\) counter - clockwise rotation about a point \(D\) means that for each point \(P(x,y)\) in the original quadrilateral, its image \(P'(x',y')\) has the property that \(D\) is the mid - point of the line segment joining \(P\) and \(P'\). Mathematically, if \(D=(a,b)\) and \(P=(x,y)\), then \(x' = 2a - x\) and \(y'=2b - y\). In a lattice (grid) - based rotation, we can also use the visual method: each vertex of the original quadrilateral is moved such that the line segment from the vertex to \(D\) is extended through \(D\) by the same length.

Step2: Analyze each option

• For option A:
• Consider the vertices of the original quadrilateral. When we rotate a quadrilateral \(180^{\circ}\) about \(D\), the relative positions of the vertices with respect to \(D\) should be reversed. In option A, the orientation and position of the vertices after the supposed rotation about \(D\) follow the \(180^{\circ}\) rotation rule. For example, if we take a vertex \(A\) (not the center \(D\)) in the original quadrilateral, its image after \(180^{\circ}\) rotation about \(D\) is placed such that \(D\) is the mid - point between \(A\) and its image.
• For option B:
• The shape and the position of the vertices do not follow the \(180^{\circ}\) rotation rule. The sides and the angles of the quadrilateral with respect to the center of rotation \(D\) are not reversed as they should be for a \(180^{\circ}\) rotation.
• For option C:
• The orientation of the quadrilateral is not consistent with a \(180^{\circ}\) rotation about \(D\). The lengths of the sides and their positions relative to \(D\) do not satisfy the \(180^{\circ}\) rotation property (where each point and its image are symmetric about \(D\)).

Answer:

A.