Question
5. If point A of a polygon is at \( (4,-8) \) what will the point A' be if the polygon is rotated \( 90^{\circ} \) counterclockwise? \( (8,4) \) \( (-4,8) \) \( (4,8) \) \( (-8,-4) \)
Ask by Crawford Page. in the United States
Mar 20,2025
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Tutor-Verified Answer
Answer
Point A' after a \(90^{\circ}\) counterclockwise rotation is \( (8, 4) \).
Solution
1. The formula for a \(90^{\circ}\) counterclockwise rotation about the origin is:
\[
(x, y) \to (-y, \,x)
\]
2. For point \( A(4, -8) \), apply the formula:
- \( x = 4 \)
- \( y = -8 \)
So,
\[
(-y, \,x) = (-(-8), \,4) = (8, \,4)
\]
3. Therefore, the rotated point \( A' \) is:
\[
(8, 4)
\]
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Simplify this solution Bonus Knowledge
When rotating a point \( (x, y) \) \( 90^{\circ} \) counterclockwise around the origin, the new coordinates become \( (-y, x) \). So for point A \( (4, -8) \), the new point A' will be \( (8, 4) \). Fun fact! The methods of rotation and reflection in geometry have been utilized since ancient times, with mathematicians like Euclid laying down the foundation. These transformations are not just for solving geometry problems; they help in computer graphics, robotics, and even navigation systems to determine object positions!
