Question

What is the new point when you rotate a geometric figure with point (5, 8)270 degrees counterclockwise about the origin?(1 point)Responses (8, 5)left parenthesis 8 comma 5 right parenthesis(−5, 8)left parenthesis negative 5 comma 8 right parenthesis(−8, −5)left parenthesis negative 8 comma negative 5 right parenthesis(8, −5)

Answer 1

To rotate a point \((x, y)\) counterclockwise about the origin by an angle \(\theta\), you can use the following formulas:

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

For a rotation of \(270^\circ\) (which is the same as \(-90^\circ\)), the cosine and sine values are:

\[
\cos(270^\circ) = 0
\]
\[
\sin(270^\circ) = -1
\]

Now, using the point \((5, 8)\):

Plugging in the values:

\[
x' = 5 \cdot 0 - 8 \cdot (-1) = 0 + 8 = 8
\]
\[
y' = 5 \cdot (-1) + 8 \cdot 0 = -5 + 0 = -5
\]

So, the new point after the rotation is \((8, -5)\).

The correct response is \((8, -5)\).