The question is which shows the pre-image of triangle x’y’z’ before the figure was rotated 90° about the origin? In geometry a pre-image is the original figure before a transformation is applied. Knowing pre-images is important in geometric transformations as it allows us to go back to the original position of figures before rotations, reflections, translations or dilations are applied.
In this post we will show you how to find the pre-image of a triangle (X’Y’Z’) that has been rotated 90 degrees counterclockwise about the origin. Follow our step by step guide and you will learn how to reverse this rotation and get the original triangle (XYZ).
Step-by-Step Guide
Step 1: Rotation Rule
When a point (x, y) is rotated 90 degrees counterclockwise about the origin it becomes (-y, x). To find the pre-image we need to reverse this process which means applying a 90 degrees clockwise rotation to each vertex of the transformed triangle (X’Y’Z’). The rule for a 90 degrees clockwise rotation is (x, y) -> (y, -x).
Step 2: Co-ordinates of the Transformed Triangle (X’Y’Z’)
Assume the co-ordinates of the transformed triangle (X’Y’Z’) are:
- X'(X1′, Y1′)
- Y'(Y1′, Y2′)
- Z'(Z1′, Z2′)
Step 3: 90 Degrees Clockwise Rotation
To find the pre-image apply the 90 degrees clockwise rotation rule to each vertex of the transformed triangle (X’Y’Z’).
Vertex X’
- Transformed coordinates: (X1′, Y1′)
- Pre-image coordinates: (Y1′, -X1′)
Vertex Y’
- Transformed coordinates: (Y1′, Y2′)
- Pre-image coordinates: (Y2′, -Y1′)
Vertex Z’
- Transformed coordinates: (Z1′, Z2′)
- Pre-image coordinates: (Z2′, -Z1′)
Step 4: Pre-Image Co-ordinates
By applying the rules the co-ordinates of the original triangle (XYZ) are:
- X(Y1′, -X1′)
- Y(Y2′, -Y1′)
- Z(Z2′, -Z1′)
Example
Given Transformed Triangle (X’Y’Z’)
- X'(2, 3)
- Y'(4, 5)
- Z'(6, 7)
90 Degrees Clockwise Rotation
- Vertex X’ (2, 3):
- Pre-image: (3, -2)
- Vertex Y’ (4, 5):
- Pre-image: (5, -4)
- Vertex Z’ (6, 7):
- Pre-image: (7, -6)
So the original triangle (XYZ) has co-ordinates:
- X(3, -2)
- Y(5, -4)
- Z(7, -6)
Visuals
Diagram 1: Original Triangle (XYZ) and Transformed Triangle (X’Y’Z’)
In the above diagram the original triangle (XYZ) is in blue and the transformed triangle (X’Y’Z’) is in red. The arrows indicate the direction of rotation.
Diagram 2: Rotation Process
This diagram shows the step by step process of rotating each vertex to find the pre-image co-ordinates.
Conclusion
Pre-images are important in geometric transformations as it helps to trace back the original figures before any transformations were applied. By following the steps above you can reverse a 90 degrees clockwise rotation to find the original triangle (XYZ) from the transformed triangle (X’Y’Z’).
For more learning visit geometric transformations and practice with different shapes and transformations to reinforce your knowledge.
FAQs
What is the formula for 90° clockwise?
Here are the rules: 90° clockwise: (x,y) becomes (y,-x) 90° counterclockwise: (x,y) becomes (-y,x) 180° clockwise and counterclockwise: (x,y) becomes (-x,-y)
How does rotation affect other geometric properties of a shape?
More formally, a rotation is a type of transformation that turns a shape around a point. We call this point the centre of rotation. A shape and its rotation have the same shape and size but will be facing a different way. A shape can be rotated clockwise or counterclockwise.
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