Question

What is the new point when you rotate a geometric figure with point (5, 8)
270 degrees counterclockwise about the origin

Answer 1

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

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

For a 270-degree counterclockwise rotation, we need to convert 270 degrees to radians:

\[
270^\circ = \frac{3\pi}{2} \text{ radians}
\]

Now, we can calculate the cosine and sine of \(270^\circ\):

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

Now, we can substitute the point \((5, 8)\) into the formulas:

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

Therefore, the new point after rotating the point \((5, 8)\) 270 degrees counterclockwise about the origin is:

\[
(8, -5)
\]