Question

Write the coordinates of the vertices after a rotation 270° counterclockwise around the origin. R is 6,3, p us 5,9, q is 10,9

Answer 1

To rotate the points \( R(6, 3) \), \( P(5, 9) \), and \( Q(10, 9) \) 270° counterclockwise around the origin, we can use the following formula for rotating a point \( (x, y) \) by an angle \( \theta \) around the origin:

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

For a 270° counterclockwise rotation, \( \theta = 270° \). Using the values of sine and cosine:

\[
\cos(270°) = 0, \quad \sin(270°) = -1
\]

Substituting in the formulas gives:

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

This means that the new coordinates after a 270° rotation will be:

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

Now we can apply this transformation to each vertex:

1. **For point \( R(6, 3) \)**:
\[
R' = (3, -6)
\]

2. **For point \( P(5, 9) \)**:
\[
P' = (9, -5)
\]

3. **For point \( Q(10, 9) \)**:
\[
Q' = (9, -10)
\]

So, the coordinates of the vertices after a 270° counterclockwise rotation around the origin are:

- \( R' = (3, -6) \)
- \( P' = (9, -5) \)
- \( Q' = (9, -10) \)