How to rotate a triangle 90 degrees clockwise?
Understand the Problem
The question is asking how to perform a geometric transformation on a triangle by rotating it 90 degrees in a clockwise direction. This involves understanding the coordinates of the triangle's vertices and applying the correct mathematical rules for rotation.
Answer
The new coordinates after rotation are $A' = (y_1, -x_1), B' = (y_2, -x_2), C' = (y_3, -x_3)$.
Answer for screen readers
The new coordinates of the triangle after a 90-degree clockwise rotation are: $$ A' = (y_1, -x_1), B' = (y_2, -x_2), C' = (y_3, -x_3) $$
Steps to Solve
- Identify the coordinates of the triangle's vertices
Start by noting the coordinates of the triangle's vertices. Let's say the vertices are $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$.
- Apply the rotation transformation
To rotate a point $(x, y)$ by 90 degrees clockwise, we can use the transformation rule: $$ (x', y') = (y, -x) $$
So, for each vertex of the triangle, apply this rule:
- For $A$: $$ A'(y_1, -x_1) $$
- For $B$: $$ B'(y_2, -x_2) $$
- For $C$: $$ C'(y_3, -x_3) $$
- Write the new coordinates
After applying the rotation rule, we obtain the new coordinates:
- The new coordinate of $A'$ becomes $(y_1, -x_1)$.
- The new coordinate of $B'$ becomes $(y_2, -x_2)$.
- The new coordinate of $C'$ becomes $(y_3, -x_3)$.
- Summarize the results
Now we can summarize the new coordinates of the rotated triangle as: $$ A' = (y_1, -x_1), B' = (y_2, -x_2), C' = (y_3, -x_3) $$
The new coordinates of the triangle after a 90-degree clockwise rotation are: $$ A' = (y_1, -x_1), B' = (y_2, -x_2), C' = (y_3, -x_3) $$
More Information
Rotating a shape in a coordinate plane is an essential concept in geometry, which helps to understand spatial transformations. Rotations are particularly useful in computer graphics, robotics, and various engineering fields.
Tips
One common mistake is applying the rotation rule incorrectly. Remember that the coordinate transformation for a 90-degree clockwise rotation is $(x, y) \to (y, -x)$ and not $(y, x)$. Double-check the signs of the coordinates.
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