Question

12 rectangle ( a ^ { prime } b ^ { prime } c ^ { prime } d ^ { prime } ) is the image of rectangle ( a b c d ) after which of the following rotations?

a ( 45 ^ { circ } ) counter clockwise rotation about origin

b ( 90 ^ { circ } ) counter clockwise about the origin

c ( 180 ^ { circ } ) rotation about the origin

d ( 90 ^ { circ } ) clockwise rotation about the origin

Explanation:

Step1: Recall rotation rules

For a point \((x,y)\) rotated \(90^{\circ}\) counter - clockwise about the origin, the new coordinates are \((-y,x)\).

Step2: Check coordinates

Let's assume a point \(A\) (say \(A=(1,1)\) in the original rectangle \(ABCD\)). After a \(90^{\circ}\) counter - clockwise rotation about the origin, using the rule \((x,y)\to(-y,x)\), it becomes \(A'=(- 1,1)\).

Step3: Eliminate other options

• A \(45^{\circ}\) rotation (option A) would not map the rectangle vertices to the given image as per standard coordinate - transformation rules.
• A \(180^{\circ}\) rotation (option C) has the rule \((x,y)\to(-x,-y)\). If \(A=(1,1)\), it would be \((-1,-1)\) which is not the case.
• A \(90^{\circ}\) clockwise rotation (option D) has the rule \((x,y)\to(y, - x)\). If \(A=(1,1)\), it would be \((1,-1)\) which is not the case.

Answer:

B. \(90^{\circ}\) counter clockwise about the origin