---
title: "Reflect triangle about y=x+2"
subject: "Computer Graphics"
chapter: 3
key: "semester_5/computer_graphics:reflect-triangle-y-x-plus-2"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/reflect-triangle-y-x-plus-2
site: BCT Engineering
---

# Reflect triangle about y=x+2

Computer Graphics · Chapter 3: Two –Dimensional Transformations

- Asked in: 2069 Asar, 6 marks
- Video: [Numerical based on reflection about line y=mx+c](https://www.youtube.com/watch?v=4bvn8wAR1_A) — sks knowledge point
- 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-y-x-plus-2
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3

## Question

Reflect a triangle whose vertices are A(-1,0), B(0,-2) and C(1, 0) about the line y = x + 2.

## Short answer

For line y = x + 2, composite transformation gives x' = y − 2, y' = x + 2. Apply to A(−1,0): (−2,1); B(0,−2): (−4,2); C(1,0): (−2,3)). The reflected triangle vertices are (−2, 1), (−4, 2), (−2, 3).

## Answer

The mirror line is y = x + 2, so m = 1 (θ = 45°) and c = 2.

**Composite transformation:** translate down by 2 (T(0,−2)), rotate by −45°, reflect about the x-axis, rotate by +45°, translate up by 2.

```text
M = T(0,2) · R(45°) · Ref_x · R(−45°) · T(0,−2)
Using the general result with m = 1, c = 2:
              | 1−m²   2m   −2mc |   | 0 2 −4 |   | 0 1 −2 |
 M = 1/(1+m²) | 2m   m²−1    2c  | = | 2 0  4 | / 2 = | 1 0  2 |
              |  0     0   1+m²  |   | 0 0  2 |   | 0 0  1 |
```
So x' = y − 2 and y' = x + 2.

**Apply to the vertices:**

```text
A(−1, 0):  x' = 0 − 2 = −2,   y' = −1 + 2 = 1     -> A'(−2, 1)
B(0, −2):  x' = −2 − 2 = −4,  y' = 0 + 2 = 2      -> B'(−4, 2)
C(1, 0):   x' = 0 − 2 = −2,   y' = 1 + 2 = 3      -> C'(−2, 3)
```

**Check:** midpoint of A and A' = (−1.5, 0.5) satisfies y = x + 2; AA' has slope (1−0)/(−2+1) = −1, perpendicular to the mirror line.

**Answer: A'(−2, 1), B'(−4, 2), C'(−2, 3).**

## Check yourself

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

1. For y = x + 2, what are m, theta and c?
   A. m = 2, θ = 45°, c = 1
   B. m = 1, θ = 45°, c = 2
   C. m = 1, θ = 90°, c = 2
   D. m = -1, θ = 135°, c = 2
   Answer: B. Slope 1 gives 45° and intercept 2. (Key point: Line parameters)

2. What is the resulting reflection mapping?
   A. x' = y, y' = x
   B. x' = y + 2, y' = x - 2
   C. x' = y - 2, y' = x + 2
   D. x' = -y - 2, y' = -x + 2
   Answer: C. The matrix [[0,1,-2],[1,0,2],[0,0,1]] gives that. (Key point: Combined mapping)

3. What is A'(from A(-1,0))?
   A. (0, 1)
   B. (2, -1)
   C. (-1, 2)
   D. (-2, 1)
   Answer: D. x' = 0 - 2 = -2, y' = -1 + 2 = 1. (Key point: Result)

4. How can a reflection be checked?
   A. The midpoint of A and A' lies on the mirror line and AA' is perpendicular to it
   B. A and A' have the same coordinates
   C. A' is the negative of A
   D. Distance from the origin is the same
   Answer: A. Midpoint on the line and perpendicular segment confirm a reflection. (Key point: Check)

## Related in Computer Graphics

- [Reflect triangle about 3X-4Y+8=0](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Areflect-triangle-3x-4y-8) — Computer Graphics, chapter 3, asked 2×
- [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
- [Scale triangle Sx=3, Sy=2](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ascaling-triangle-sx3-sy2) — Computer Graphics, chapter 3

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