The General Theory of Rotation and Enlargement
ByShlok Ghop
SECTION A: Rotation
To master rotation means to visualize.
For e.g:-
Lets say we want to rotate shape A 90 clockwise with center(6,5)
Now there are two ways to solve this:
1. ChatGPT Way:
To rotate Shape A with coordinates (2,5)(2, 5)(2,5), (2,2)(2, 2)(2,2), (4,4)(4, 4)(4,4),
and (4,2)(4, 2)(4,2) 90 degrees clockwise around the center (6,5)(6, 5)(6,5), follow these
steps:
1. Translate the shape so the center of rotation becomes the origin:
o
Subtract (6,5)(6, 5)(6,5) from each point.
2. Apply the 90-degree clockwise rotation formula:
o
For a point (x,y)(x, y)(x,y), the new coordinates after a 90-degree clockwise
rotation around the origin are (y,−x)(y, -x)(y,−x).
3. **Translate the shape back by adding (6,5)(6, 5)(6,5) to each point.
Step-by-Step Calculation:
1. Subtract (6, 5) from each point:

(2,5)→(2−6,5−5)=(−4,0)(2, 5) \rightarrow (2 - 6, 5 - 5) = (-4,
0)(2,5)→(2−6,5−5)=(−4,0)

(2,2)→(2−6,2−5)=(−4,−3)(2, 2) \rightarrow (2 - 6, 2 - 5) = (-4, 3)(2,2)→(2−6,2−5)=(−4,−3)

(4,4)→(4−6,4−5)=(−2,−1)(4, 4) \rightarrow (4 - 6, 4 - 5) = (-2, 1)(4,4)→(4−6,4−5)=(−2,−1)

(4,2)→(4−6,2−5)=(−2,−3)(4, 2) \rightarrow (4 - 6, 2 - 5) = (-2, 3)(4,2)→(4−6,2−5)=(−2,−3)
2. Apply the 90-degree clockwise rotation formula:

(−4,0)→(0,4)(-4, 0) \rightarrow (0, 4)(−4,0)→(0,4)

(−4,−3)→(−3,4)(-4, -3) \rightarrow (-3, 4)(−4,−3)→(−3,4)

(−2,−1)→(−1,2)(-2, -1) \rightarrow (-1, 2)(−2,−1)→(−1,2)

(−2,−3)→(−3,2)(-2, -3) \rightarrow (-3, 2)(−2,−3)→(−3,2)
3. Add (6, 5) to each point:

(0,4)→(0+6,4+5)=(6,9)(0, 4) \rightarrow (0 + 6, 4 + 5) = (6, 9)(0,4)→(0+6,4+5)=(6,9)

(−3,4)→(−3+6,4+5)=(3,9)(-3, 4) \rightarrow (-3 + 6, 4 + 5) = (3,
9)(−3,4)→(−3+6,4+5)=(3,9)

(−1,2)→(−1+6,2+5)=(5,7)(-1, 2) \rightarrow (-1 + 6, 2 + 5) = (5,
7)(−1,2)→(−1+6,2+5)=(5,7)

(−3,2)→(−3+6,2+5)=(3,7)(-3, 2) \rightarrow (-3 + 6, 2 + 5) = (3,
7)(−3,2)→(−3+6,2+5)=(3,7)
Result:
The new coordinates of Shape A after a 90-degree clockwise rotation around the point
(6,5)(6, 5)(6,5) are:

(6,9)(6, 9)(6,9)

(3,9)(3, 9)(3,9)

(5,7)(5, 7)(5,7)

(3,7)(3, 7)(3,7)
These are the vertices of the rotated shape.
I know you haven’t even tried reading this
2. Visualization
For now imagine the shape in this manner
Now lets come back to the graph
Now l have labelled one of the vertices of shape A as X
A 90 degree is a right angle. To rotate Shape A we will have to perform the number of difference in
point X and center on the center and a perpendicular line
I will explain
The line which I have made forms a
90 degree in clockwise direction
from Shape A. Point X is 4 units left
to the center. The newly formed
point(x1) and X should form a 90
degree.
By going 4 units up, I form X1 which
gives me the first vertices.
Another point, Y, if we move 2
blocks right and 1 block up: we
reach the centre. We can show this
2
in the form of vector ( )
1
−1
Now flip the vector into ( )
2
Why?
Because we need to form a 90 degree
This time perform the down part first then the upper one
2.1 Vectors
Or in an easier way you could say that:
You must inverse the steps when looking at the perspective of point Y
Again visualize
In the inverse vector of (1, -2)
-2 becomes positive
And 1 becomes negative
−1
Making ( )
2
Same for point X
If you look from the prospective of X, the center is 4 units down. In the form of vector it is (
0
)
−4
0
Inverse it and it becomes ( )
4
And perform that from the center and will for sure get X1.
Do it for all the points and you will get shape A rotated to 90 degrees clockwise.
This might seem much but when put into practice, its way easier and as you continue using this
method, you will improve your visualization skills as well