---
title: "Halve rectangle ABCD about its centre"
subject: "Computer Graphics"
chapter: 3
key: "semester_5/computer_graphics:scale-rectangle-half-size"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/scale-rectangle-half-size
site: BCT Engineering
---

# Halve rectangle ABCD about its centre

Computer Graphics · Chapter 3: Two –Dimensional Transformations

- Asked in: 2072 Kartik, 10 marks
- Video: [6.9- Numerical- Homogeneous Coordinate - Composition Of 2d Transformation In Computer Graphics Hindi](https://www.youtube.com/watch?v=59Wg2VMFMs8) — TutorialsSpace- Er. Deepak Garg
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/scale-rectangle-half-size
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3

## Question

Find the transformation matrix that transforms a rectangle ABCD whose center is at (4,2) so that it is reduced to half of its size, the center remaining the same. The co-ordinates of ABCD are A(0,0), B(0,4), C(8,4) and D(8,0). Find the coordinates of the new square. Also derive the transformation matrix to convert this rectangle to square.

## Short answer

Scale about fixed point (4,2) using M with sx=sy=0.5: corners become (2,1), (2,3), (6,3), (6,1). For a half-size square, use sx=0.25, sy=0.5 giving (3,1), (3,3), (5,3), (5,1). Center unchanged.

## Answer

**Given:** rectangle A(0,0), B(0,4), C(8,4), D(8,0), centre (4,2). The scaling must keep the centre fixed, so it is done about the fixed point (4,2):

M = T(4,2) · S(sx,sy) · T(−4,−2)

```text
      | sx  0   4(1−sx) |
 M =  | 0   sy  2(1−sy) |
      | 0   0      1    |
```

#### (a) Reduce to half of its size (sx = sy = 0.5)
```text
      | 0.5  0   2 |
 M =  |  0  0.5  1 |       x' = 0.5x + 2,  y' = 0.5y + 1
      |  0   0   1 |
A(0,0) → (2,1)    B(0,4) → (2,3)    C(8,4) → (6,3)    D(8,0) → (6,1)
```
The new rectangle is 4 × 2 with the same centre (4,2).

#### (b) Convert the rectangle to a square (centre unchanged)
The rectangle is 8 wide and 4 high. To get a square of side 4, shrink x by half and keep y: sx = 0.5, sy = 1.

```text
      | 0.5  0   2 |
 M =  |  0   1   0 |
      |  0   0   1 |
A → (2,0)   B → (2,4)   C → (6,4)   D → (6,0)
```
This is a square of side 4. (Instead, sx = 1, sy = 2 gives a square of side 8.)

#### (c) Both: half-size square
Reducing to half size and making it a square (side 2) needs sx = 0.25, sy = 0.5:

```text
      | 0.25  0   3 |
 M =  |  0   0.5  1 |
      |  0    0   1 |
A → (3,1)   B → (3,3)   C → (5,3)   D → (5,1)
```

**Answer:** the half-size rectangle has corners (2,1), (2,3), (6,3), (6,1); the half-size square has corners (3,1), (3,3), (5,3), (5,1); all with centre (4,2).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is the center kept fixed?
   A. M = T(4,2)·S·T(-4,-2)
   B. M = S only
   C. M = T(-4,-2)·S·T(4,2)
   D. M = S·R
   Answer: A. Move the center to the origin, scale, move back. (Key point: Fixed-point scaling)

2. What is the matrix for sx = sy = 0.5 about (4,2)?
   A. [[0.5,0,4],[0,0.5,2],[0,0,1]]
   B. [[0.5,0,2],[0,0.5,1],[0,0,1]]
   C. [[0.5,0,0],[0,0.5,0],[0,0,1]]
   D. [[2,0,-2],[0,2,-1],[0,0,1]]
   Answer: B. Translation parts are 4(1-0.5) = 2 and 2(1-0.5) = 1. (Key point: Half size)

3. To change the 8 by 4 rectangle into a square of side 4, what factors?
   A. sx = 1, sy = 0.5
   B. sx = 0.25, sy = 0.5
   C. sx = 0.5, sy = 1
   D. sx = 2, sy = 1
   Answer: C. Width 8 halves to 4 and height stays 4. (Key point: Square)

4. What are the half-size rectangle corners?
   A. (0,0), (0,2), (4,2), (4,0)
   B. (3,1), (3,3), (5,3), (5,1)
   C. (2,0), (2,4), (6,4), (6,0)
   D. (2,1), (2,3), (6,3), (6,1)
   Answer: D. A(0,0)->(2,1) etc., a 4 by 2 rectangle. (Key point: Result)

## Related in Computer Graphics

- [Rectangle window to viewport mapping](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aviewing-transform-rectangle-window-viewport) — Computer Graphics, chapter 3
- [Derive perspective projection matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aperspective-projection-matrix-derive) — Computer Graphics, chapter 4, asked 3×
- [Derive parallel projection matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aparallel-projection-matrix-derive) — Computer Graphics, chapter 4, asked 2×
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- [Square to rhombus; reflection about y=x](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Asquare-to-rhombus-reflection-y-x) — Computer Graphics, chapter 3
- [3D window to viewport transformation](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Awindow-viewport-3d-matrix) — Computer Graphics, chapter 4

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