Question

10.4.2 quiz: transformations
a. 360°
b. 180°
c. 90°
d. 270°

Explanation:

Step1: Analyze the transformation

When a figure is rotated \(180^{\circ}\) about the origin, the rule for a point \((x,y)\) is \((x,y)\to(-x,-y)\).

Step2: Check the coordinates

Suppose a vertex of the black square (pre - image) has coordinates \((x,y)\). For example, if we consider a general vertex \(A(0,0)\) (assuming \(A\) is at the origin - like position in the coordinate system for simplicity of rotation concept). After a \(180^{\circ}\) rotation, it will go to \((0,0)\) (invariant point for rotation about the origin), but for non - origin points. If we take a point \((a,b)\) in the black square, its image in the red square (post - image) will satisfy the \(180^{\circ}\) rotation rule \((a,b)\to(-a,-b)\). Visually, the red square is the opposite of the black square with respect to the origin (center of rotation conceptually), which is characteristic of a \(180^{\circ}\) rotation. A \(90^{\circ}\) rotation would change the orientation in a different way (e.g., if rotated \(90^{\circ}\) clockwise, \((x,y)\to(y, - x)\)), a \(270^{\circ}\) rotation \((x,y)\to(-y,x)\) and a \(360^{\circ}\) rotation would map the figure onto itself exactly (coincide with the pre - image). So the rotation angle is \(180^{\circ}\).

Answer:

B. \(180^{\circ}\)