## Key Concepts

- Understand about reflection.
- Graph a reflection in horizontal and vertical lines.
- Graph a reflection y = x
- Coordinate rules of reflection
- Graph a reflection y = –x

## Graphical Reflections

### Introduction

**What transformation is shown in the below image?**

**Answer:**

**Reflection along Horizontal line**

**What transformation is shown in the below image?**

**Answer:**

**Reflection along Horizontal line **

**Which image shows reflection transformation?**

**Answer: c**

### Reflection

A reflection is a transformation that uses a line to reflect an image, which is similar to a mirror. This mirror line is called the reflection line or line of reflection.

A reflection in a line m maps every point Q in the plane to a point Q’, so that for each point one of the following properties is true:

- If Q is not on n, then n is the perpendicular bisector of QQ’−QQ′- , or
- If Q is on n, then Q = Q’.

### Graph reflections along horizontal and vertical lines

Let us understand this concept with the help of an examples.

**Example 1:**

The vertices of ∆ABC are A (3, 5), B (2, 2), and C (5, 4). Graph the reflection of ∆ABC, in the line n: x = 3

**Solution:**

Point A is on the n, so its reflection A’ is also on the n at A’ (3, 5)

Point B is 1 unit left to the n, so its reflection is 1 unit to the right of n at B’ (4, 2)

Point C is 2 units right to n, so its reflection is 2 units to the right of n at C’ (1, 4).

**Example 2:**

The vertices of ∆ABC are A (3, 5), B (2, 2), and C (5, 4). Graph the reflection of ΔABC, in the line m: y = 2

**Solution:**

Point A is 3 units above m, so its reflection is 3 units below m at (3, -1)

Point B is on the m, so its reflection B’ is on m.

Point C is 2 units above m, so its reflection is 2 units to the below m at C’ (5, 0).

### Graph a reflection in y = x

Let us understand this concept with the help of an example:

**Example 1:**

The endpoints of A are A (–1, 2) and B (1, 2). Reflect the segment in the line y = x. Graph the segment and its image.

**Solution:**

The slope of y = x is 1. The segment from A to its image, AA′-

is perpendicular to the line of reflection y = x, so the slope of AA′- will be –1.

From A, move 1.5 units right and 1.5 units down to y = x. From that point, move 1.5 units right and 1.5 units down to locate A’ (3, –1).

The slope of BB′- will also be –1. From B, move 0.5 units right and 0.5 units down to y = x. Then move 0.5 units right and 0.5 units down to locate B’ (2, 1).

**Coordinate rules for reflection:**

- If (a, b) is reflected in the x-axis, its image is the point (a, –b).
- If (a, b) is reflected in the y-axis, its image is the point (–a, b).
- If (a, b) is reflected in the line y = x, its image is the point (b, a).
- If (a, b) is reflected in the line y = –x, its image is the point (–b, –a).

### Graph reflection for y = -x

Let us understand this concept with the help of an example:

**Example 1:**

Reflect AB- in the line y = –x. Graph AB- and its image. A (–1, 2) and B (1, 2).

**Solution:**

Use the coordinate rule for reflecting in y = –x

(a, b) → (–b, –a)

A (–1, 2) → (–2, 1)

B (1, 2) → (–2, –1)

**Example 2:**

Reflect AB- in the line y = –x. Graph AB and its image. A (–3, 3) and B (–8, 2).

**Solution:**

Use the coordinate rule for reflecting in y = –x

(a, b) → (–b, –a)

A (–3, 2) → (–2, 3)

B (–8, 2) → (–2, 8)

**Examples:**

1. Reflect ∆ABC are A (–3, 2), B (–4, 5), and C (–1, 6) in the line y = 3. Graph ΔABC and its image

**Solution:**

Point A is 1 unit below y, so its reflection is 1 unit above y at

A’ (–3, 4)

Point B is 2 units above y, so its reflection is 2 units below y at

B (–4, 1)

Point C is 3 units above y, so its reflection is 3 units to the below y at C’ (–1, 0).

2. The endpoints of PQ−PQ- are P (4, –2) and Q(9, –2). Reflect the segment in the line y = x. Graph the segment and its image.

**Solution:**

If (a, b) is reflected in the line y = x, its image is the point (b, a).

(a, b) → (b, a)

P (4, –2) → (–2,4)

Q (9, –2) → (–2,9)

## Exercise

- The vertices of ΔABC are A (3, 5), B (2, 2), and C (5, 4). Graph the reflection of ΔABC, in the line n: y =4
- The vertices of ΔABC are A (3, 5), B (2, 2), and C (5, 4). Graph the reflection of ΔABC, in the line n: x =2
- Graph ΔABC with vertices A (1, 3), B (4, 4), and C (3, 1). Reflect ΔABC in the line y = –x. Graph image.

Graph ΔABC with vertices A (1, 3), B (4, 4), and C (3, 1)

Reflect ΔABC in the line y = x. Graph image.

The vertices of ΔABC are A (3, 5), B (2, 2), and C (5, 4). Graph the reflection of ΔABC, in the line m: y = 2 Graph the reflection of the polygon in the given line.

- Graph the reflection of the polygon in the given line.

- Graph the reflection of the polygon in the given line.

- Graph the reflection of the polygon in the given line.

- What is the line of reflection for ABC and its image?

a. y= 0 b. y = –x c. x=1 d. y=x

### Concept Map

### What we have learned

- Understand about reflection.
- Graph a reflection in horizontal and vertical lines.
- Graph a reflection y = x
- Coordinate rules of reflection
- Graph a reflection y = –x

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