Question

quadrilateral abcd is rotated ( 270^{circ} ) clockwise about the origin. what is the orientation of the rotated figure, and what quadrant does it lie in?
a. fourth quadrant
b. fourth quadrant
c. second quadrant
d. first quadrant

Explanation:

Step1: Understand rotation rules

A \(270^{\circ}\) clockwise rotation about the origin is equivalent to a \(90^{\circ}\) counter - clockwise rotation. When a figure is rotated \(270^{\circ}\) clockwise about the origin, the orientation of the vertices changes. The order of the vertices (orientation) of the quadrilateral ABCD (original) will be reversed in a sense. For example, if the original order of vertices is \(A - B - C - D\), after rotation, the order of vertices in the image \(A'B'C'D'\) will be such that the shape is rotated.

Step2: Analyze quadrant location

The original quadrilateral ABCD is in the third quadrant. A \(270^{\circ}\) clockwise rotation:
Let \((x,y)\) be a point in the original figure. The rule for a \(270^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y, - x)\). Points in the third quadrant have \(x<0\) and \(y < 0\). After rotation \((x,y)\to(y,-x)\), if \(x=-a,a>0\) and \(y = - b,b>0\), then the new coordinates are \((-b,a)\) which lie in the second quadrant.

Step3: Analyze orientation

When we rotate a figure \(270^{\circ}\) clockwise (or \(90^{\circ}\) counter - clockwise), the orientation of the vertices (the order in which we traverse the vertices of the quadrilateral) is reversed. If we assume the original orientation of the quadrilateral ABCD (traversing \(A - B - C - D\)), after a \(270^{\circ}\) clockwise rotation, the orientation of \(A'B'C'D'\) will be such that if we start from \(A'\), the order of vertices will be \(A' - D' - C' - B'\) (reversed orientation compared to the original in terms of the direction of traversal).

Answer:

C.