Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 136

Transformations (Rotations)

When rotating a shape, instead of using tracing paper to physically rotate the shape, can we use maths to work out the co-ordinates of where the shape will be rotated to?

I'm talking about GCSE maths, so, fairly basic algebra perhaps, maybe vectors too?

Also, when we are shown the rotation and asked to describe it, rather than using trial and error (with tracing paper) to find the co-ordinates about which the the shape has been rotated can we use maths instead to work it out?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Transformations (Rotations)

hi paulb203

If the origin is the centre of rotation, then you can write the coordinates as 2 by 1 vectors and multiply by a 2 D rotation matrix.  Here's an example for  rotation of angle alpha around (0,0) points (3,2) and (4,5)

This moves (3,2) to (3cos alpha - 2sin alpha, 3sin alpha + 2 cos alpha)

Matrices cannot be used if the origin isn't the centre.

If you know that the line AB is rotated onto A'B' then you can use a construction to get the centre.

Join AA' and find the midpoint. Construct the perpendicular bisector of AA'. Similarly construct the perpendicular bisector of BB'.

Where the two bisectors cross is the centre.

See http://www.mathisfunforum.com/viewtopic.php?id=22507

If you post a specific question I'll demonstrate.

Later edit:

Suppose the centre is at (c,d) and a point (x,y) is rotated 90 anticlockwise.

I'll put the vectors as 1 by 2 to save time with vertical arrays.

The vector from the centre to the point is (x-c    y-d). 

So a vector of the same length through the centre and at right angles is (-y + d     x - c)

So you can apply that vector to the centre to find where the point is rotated to.

eg. centre (6,2) point (10,5)    Then the vector from the centre is (10 - 6     5 - 2) = (4   3)

A vector at right angles to this and with the same length is ( -3   4)   **

So starting at the centre and going (-3    +4) takes us to (6 -3    2 + 4) = (3 , 6)

If you try plotting these points you'll see it works.

** (3 -4) also works but gives a clockwise rotation of 90.

This method will allow you to rotate about any point, but only 90, 180, 270.  I doubt GCSE would set a question where a more difficult angle is used.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 136

Re: Transformations (Rotations)

Thanks, Bob.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Transformations (Rotations)

Are you sitting the GCSE exam soon?  What board?  I have spent a major part of my career teaching this and now I'm retired.  I'm very happy to help if I can so do please post back some more.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Transformations (Rotations)

When rotating a shape, instead of using tracing paper to physically rotate the shape, can we use maths to work out the co-ordinates of where the shape will be rotated to?

I'm talking about GCSE maths, so, fairly basic algebra perhaps, maybe vectors too?

Also, when we are shown the rotation and asked to describe it, rather than using trial and error (with tracing paper) to find the co-ordinates about which the the shape has been rotated can we use maths instead to work it out?

What does GCSE stand for?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Transformations (Rotations)

General Certificate of Secondary Education. It's a UK exam mainly taken by 16 year olds.

B

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Transformations (Rotations)

General Certificate of Secondary Education. It's a UK exam mainly taken by 16 year olds.

B

Really? Impressive to know that teenagers in the UK are advanced in terms of mathematics. Student in the United States, particularly in BLUE STATES don't know what planet they live on.