Demonstrating the properties of rotations, if a line segment with endpoints (0,−3)

Question

and (0,−7)
is rotated 90°
clockwise, what is an endpoint of this rotated segment?(1 point)
Responses

(−3, 0)
left parenthesis negative 3 comma 0 right parenthesis

(0, 3)
left parenthesis 0 comma 3 right parenthesis

(0, 7)
left parenthesis 0 comma 7 right parenthesis

(3, 0)

Answer 1

To rotate a point \( (x, y) \) 90° clockwise around the origin, you can use the rotation formula:

\[ (x', y') = (y, -x) \]

Now, let's apply this to the endpoint \( (0, -3) \):

1. Original point: \( (0, -3) \)
2. Applying the formula: \[ x' = y = -3, \quad y' = -x = -0 = 0 \] Therefore, the new coordinates after rotation are \( (-3, 0) \).

Next, we can apply the same process to the second endpoint \( (0, -7) \):

1. Original point: \( (0, -7) \)
2. Applying the formula: \[ x' = y = -7, \quad y' = -x = -0 = 0 \] Therefore, the new coordinates after rotation are \( (-7, 0) \).

So, after rotating the segment with endpoints \( (0, -3) \) and \( (0, -7) \) 90° clockwise, the new endpoints become \( (-3, 0) \) and \( (-7, 0) \).

Thus, one endpoint of the rotated segment is:

\[ \text{(−3, 0)} \]

So the answer is \((−3, 0)\).

Answer 2

Rotate a point on a line at (−4, 6)
180°
counterclockwise. What is the resulting point?(1 point)
Responses

(−4, −6)
left parenthesis negative 4 comma negative 6 right parenthesis

(−6, 4)
left parenthesis negative 6 comma 4 right parenthesis

(6, −4)
left parenthesis 6 comma negative 4 right parenthesis

(4, −6)