Question

1. a point p has coordinates (2,3). what are the coordinates of the image of p after a reflection across the line y = 1, followed by a 90 - degree counterclockwise rotation about the origin, and then a translation of (x,y)→(x - 2,y + 1)?

• a) (-1,4)
• b) (-1,3)
• c) (-4,2)
• d) (0,1)

a) (-1,4)
b) (-1,3)
c) (-4,2)
d) (0,1)

Explanation:

Step1: Reflection across y = 1

The distance between y - coordinate of point P(2,3) and y = 1 is \(3 - 1=2\). After reflection across y = 1, the y - coordinate becomes \(1-(3 - 1)= - 1\) and the x - coordinate remains the same. So the point after reflection is \((2,-1)\).

Step2: 90 - degree counter - clockwise rotation about the origin

The rule for a 90 - degree counter - clockwise rotation about the origin \((x,y)\to(-y,x)\). For the point \((2,-1)\), after rotation, it becomes \((1,2)\).

Step3: Translation \((x,y)\to(x - 2,y + 1)\)

For the point \((1,2)\), \(x=1,y = 2\). After translation, \(x=1-2=-1\) and \(y=2 + 1=3\). The final point is \((-1,3)\).

Answer:

B. (-1,3)