Question

1. what happens to the coordinates of a point ((x, y)) after a 90-degree counterclockwise rotation around the origin?

a. ((y, -x))

b. ((-y, -x))

c. ((-y, x))

d. ((x, -y))

Explanation:

To determine the coordinates of a point \((x, y)\) after a 90 - degree counterclockwise rotation around the origin, we can use the rule for 90 - degree counterclockwise rotations. The general rule is that a point \((x,y)\) becomes \((-y,x)\) after a 90 - degree counterclockwise rotation about the origin. Let's verify with an example. Take the point \((1,0)\). A 90 - degree counterclockwise rotation around the origin should take it to \((0,1)\). Using the rule \((-y,x)\) where \(x = 1\) and \(y=0\), we get \((-0,1)=(0,1)\), which is correct. Another example: take \((0,1)\). A 90 - degree counterclockwise rotation should take it to \((-1,0)\). Using the rule, \(x = 0\), \(y = 1\), so \((-1,0)\), which is correct. Now let's check the other options:

• Option a: \((y,-x)\) is the rule for a 90 - degree clockwise rotation, not counterclockwise.
• Option b: \((-y,-x)\) is not the rule for a 90 - degree counterclockwise rotation.
• Option d: \((x,-y)\) is the rule for a 180 - degree rotation? No, actually \((x,-y)\) is the rule for a reflection over the x - axis, and the rule for 180 - degree rotation is \((-x,-y)\).

Answer:

c. \((-y, x)\)