EDEXCEL GCSE HIGHER MATHS - What are the Invariant Points and Their Applications

Invariant Points in GCSE Higher Maths

What are Invariant Points?

Invariant points are points that remain unchanged when a transformation is applied. In GCSE Higher Maths, we focus on invariant points under transformations like:

• Reflections: A point is invariant under a reflection if it lies on the line of reflection.
• Rotations: A point is invariant under a rotation if it is the centre of rotation.
• Translations: No points are invariant under a translation.
• Enlargements: A point is invariant under an enlargement if it is the centre of enlargement.

Finding Invariant Points

To find invariant points, we can use the following steps:

1. Identify the transformation: Determine the type of transformation being applied.
2. Apply the transformation: Apply the transformation to a general point (x, y).
3. Set the original and transformed coordinates equal: Set the original coordinates (x, y) equal to the transformed coordinates.
4. Solve the equations: Solve the resulting equations to find the values of x and y, which will give you the invariant point(s).

Applications of Invariant Points

Invariant points are useful in various mathematical concepts:

• Finding the equation of a line of reflection: The midpoint of the line segment connecting a point and its image under reflection is the invariant point (on the line of reflection).
• Finding the centre of rotation: The invariant point under a rotation is the centre of rotation.
• Finding the centre of enlargement: The invariant point under an enlargement is the centre of enlargement.
• Determining the scale factor of an enlargement: By comparing the distance between a point and its image to the distance between the point and the centre of enlargement, you can find the scale factor.

Examples

Example 1: Reflection

Question: Find the invariant point(s) under the reflection in the line y = x.

Solution:

1. Transformation: Reflection in the line y = x.
2. Apply transformation: The reflection of a point (x, y) in the line y = x is (y, x).
3. Set coordinates equal: (x, y) = (y, x)
4. Solve equations: x = y.

Therefore, the invariant points are all points that lie on the line y = x.

Example 2: Rotation

Question: Find the invariant point(s) under the rotation of 90 degrees anticlockwise about the point (2, 3).

Solution:

1. Transformation: Rotation of 90 degrees anticlockwise about the point (2, 3).
2. Apply transformation: The rotation of a point (x, y) about (2, 3) by 90 degrees anticlockwise is (y - 3 + 2, -(x - 2) + 3).
3. Set coordinates equal: (x, y) = (y - 3 + 2, -(x - 2) + 3)
4. Solve equations:
5. x = y - 1
6. y = -x + 5
7. Solving these equations, we get x = 2 and y = 3.

Therefore, the invariant point is (2, 3).

Conclusion

Invariant points provide a useful tool for understanding and solving problems related to transformations in GCSE Higher Maths. By understanding how to find and use invariant points, you can gain a deeper understanding of these transformations and their applications in various mathematical contexts.