Question

Given point A (2, 4)
A

(
2
,

4
)
on the triangle Quadrant I, where is the new point if the triangle is rotated 270 degrees counterclockwise?

(1 point)
Responses

(4,−2)
(
4
,
−
2
)
left parenthesis 4 comma negative 2 right parenthesis

(−4, 2)
(
−
4
,

2
)
left parenthesis negative 4 comma 2 right parenthesis

(−4,−2)
(
−
4
,
−
2
)
left parenthesis negative 4 comma negative 2 right parenthesis

(−2,−4)

Answer 1

To find the new coordinates of point A(2, 4) after a 270-degree counterclockwise rotation around the origin, we can use the rotation formula.

The formula for rotating a point \((x, y)\) by an angle \(\theta\) around the origin is given by:

\[
(x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)
\]

For a 270-degree rotation, \(\theta = 270^\circ\) or \(\theta = \frac{3\pi}{2}\) radians.

Using the values of sine and cosine for 270 degrees:
- \(\cos(270^\circ) = 0\)
- \(\sin(270^\circ) = -1\)

Now, we can substitute these values into the rotation formula:

\[
(x', y') = \left(2 \cdot 0 - 4 \cdot (-1), 2 \cdot (-1) + 4 \cdot 0\right)
\]

Calculating the new coordinates:

1. For \(x'\):
\[
x' = 0 + 4 = 4
\]

2. For \(y'\):
\[
y' = -2 + 0 = -2
\]

Thus, the new coordinates after rotating point A(2, 4) by 270 degrees counterclockwise around the origin are \((4, -2)\).

Therefore, the correct answer is:

**(4, -2)**