Reflections, rotations and enlargements · Foundation
| English | 中文 | Pinyin |
|---|---|---|
| centre of enlargement/ˈsentə ɒv enˈlɑːdʒmənt/ | 位似中心 | wèi shì zhōng xīn |
Where is the enlargement centre?
- A logo is enlarged around a point away from the origin. Multiplying its coordinates alone puts it in the wrong place.
- This lesson studies centre of enlargement 位似中心: The point from which each point's displacement is multiplied by a scale factor.
Choose the mathematical structure
- A translation adds a vector. A reflection reverses signed perpendicular distance from a mirror line. A rotation needs a centre, angle and direction. For enlargement from C, use new P=C+k(P-C).
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines centre of enlargement?
The point from which each point's displacement is multiplied by a scale factor.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For centre C=(1,1), point P=(3,2) and factor 2, the image is (5,3). At factor 1/2 the image is C+(1/2)(P-C)=(2,1.5), halfway from C to P. Reflecting (3,2) in the y-axis gives (-3,2). Rotating it 90° anticlockwise about the origin gives (-2,3). Translation by (2,-1) gives (5,1). State the centre, line or vector as appropriate.
Reflections, rotations and enlargements
A translation adds a vector
Explain why scaling P directly would give a different enlargement.
For C=(1,1),P=(3,2),k=2, find the image x-coordinate.
Image=C+2(P-C). Its x-coordinate is 1+2(3-1)=5.
Test a tempting shortcut
- Specify a reflection line, rotation centre/angle/direction or enlargement centre/factor. Fractional positive enlargement factors reduce a shape without reversing its position about the centre.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every enlargement is centred at the origin unless its scale factor is negative. This claim is false. Explain which definition or assumption it violates.
For that enlargement, find the image y-coordinate.
Its y-coordinate is 1+2(2-1)=3.
Every enlargement is centred at the origin unless its scale factor is negative.
Specify a reflection line, rotation centre/angle/direction or enlargement centre/factor. Fractional positive enlargement factors reduce a shape without reversing its position about the centre.
Interpret a new situation
- AQA G7 Foundation includes single reflections, rotations, translations and enlargements with positive integer/fractional factors. Negative factors and combinations of isometries belong to separate Higher teaching.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Reflect (3,2) in the y-axis. Find the new x-coordinate.
Reflection in the vertical axis negates x: 3 becomes -3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The point from which each point's displacement is multiplied by a scale factor. Choose the relationship, show the method, check its assumptions and interpret the result.