---
title: "Reflect triangle about 3X-4Y+8=0"
subject: "Computer Graphics"
chapter: 3
key: "semester_5/computer_graphics:reflect-triangle-3x-4y-8"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/reflect-triangle-3x-4y-8
site: BCT Engineering
---

# Reflect triangle about 3X-4Y+8=0

Computer Graphics · Chapter 3: Two –Dimensional Transformations

- Asked in: 2078 Kartik, 8 marks; 2075 Asoj, 8 marks (asked 2 times)
- Video: [Reflection about a line in Computer Graphics // numerical on reflection about a given line](https://www.youtube.com/watch?v=T3ewDqW0qjs) — Yachana Bhawsar
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/reflect-triangle-3x-4y-8
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3

## Question

Reflect the triangle ABC about the line 3X - 4Y + 8 = 0. The position vectors of the coordinates of ABC are A(4, 1), B(5, 2) and C(4, 3).

## Short answer

For line 3x-4y+8=0, m=3/4, c=2, composite matrix is [[0.28,0.96,-1.92],[0.96,-0.28,2.56],[0,0,1]]. Applying it gives A'(0.16,6.12), B'(1.40,6.80), C'(2.08,5.56).

## Answer

**Step 1: find the slope and intercept.** 3X − 4Y + 8 = 0 gives Y = (3/4)X + 2, so m = 3/4 = 0.75, c = 2, and θ = tan⁻¹(0.75) = 36.87° (cosθ = 0.8, sinθ = 0.6).

**Step 2: composite transformation** M = T(0,c) · R(θ) · Ref_x · R(−θ) · T(0,−c), which simplifies to

```text
              | 1−m²    2m     −2mc  |
 M = 1/(1+m²) | 2m     m²−1     2c   |
              |  0       0     1+m²  |
```

With m = 0.75: 1 + m² = 1.5625, 1 − m² = 0.4375, 2m = 1.5, 2mc = 3, 2c = 4. Also cos2θ = 0.28, sin2θ = 0.96.

```text
        | 0.28   0.96  −1.92 |
   M =  | 0.96  −0.28   2.56 |
        |  0      0      1   |
```

**Step 3: apply to the vertices** (homogeneous column vectors [x y 1]ᵀ):

```text
A(4,1):  x' = 0.28·4 + 0.96·1 − 1.92 = 1.12 + 0.96 − 1.92 = 0.16
         y' = 0.96·4 − 0.28·1 + 2.56 = 3.84 − 0.28 + 2.56 = 6.12
B(5,2):  x' = 1.40 + 1.92 − 1.92 = 1.40
         y' = 4.80 − 0.56 + 2.56 = 6.80
C(4,3):  x' = 1.12 + 2.88 − 1.92 = 2.08
         y' = 3.84 − 0.84 + 2.56 = 5.56
```

**Check (for A):** distance of A from the line = (3·4 − 4·1 + 8)/5 = 3.2; A' = A − 2·3.2·(3,−4)/5 = (4 − 3.84, 1 + 5.12) = (0.16, 6.12). It agrees.

**Answer: A'(0.16, 6.12), B'(1.4, 6.8), C'(2.08, 5.56).**

## Check yourself

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

1. Rewriting 3X - 4Y + 8 = 0 as Y = mX + c, what are m and c?
   A. m = 4/3, c = 2
   B. m = 3/4, c = 2
   C. m = 3/4, c = -2
   D. m = -3/4, c = 8
   Answer: B. Solving for Y gives Y = (3/4)X + 2, so m = 0.75 and c = 2. (Key point: Slope and intercept)

2. Which sequence reflects a point about a line Y = mX + c?
   A. Reflect about the x-axis, then translate by (0,c)
   B. Translate by (0,-c), rotate by -theta, reflect about the x-axis, rotate by +theta, translate by (0,c)
   C. Rotate by theta, reflect about the y-axis, rotate back
   D. Scale by m, then translate by c
   Answer: B. The line is brought onto the x-axis, reflected, and then moved back. (Key point: Composite transformation)

3. With m = 0.75, what is the top-left entry (1-m^2)/(1+m^2) of the reflection matrix?
   A. 0.96
   B. 0.6
   C. 0.28
   D. 0.8
   Answer: C. (1 - 0.5625)/1.5625 = 0.28, which equals cos2theta. (Key point: Matrix entries)

4. What is the reflected image A' of A(4,1)?
   A. (1.12, 3.84)
   B. (6.12, 0.16)
   C. (0.16, 6.12)
   D. (-0.16, 6.12)
   Answer: C. x' = 0.28*4 + 0.96*1 - 1.92 = 0.16 and y' = 0.96*4 - 0.28 + 2.56 = 6.12. (Key point: Result)

## Related in Computer Graphics

- [Reflect triangle about y=x+2](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Areflect-triangle-y-x-plus-2) — Computer Graphics, chapter 3
- [Visibility of triangle ABC from (5,5,5)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aback-face-triangle-abc-observer) — Computer Graphics, chapter 7
- [Reflect object about y=x+1](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Areflect-object-y-x-plus-1) — Computer Graphics, chapter 3

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