#### Need Help?

Get in touch with us

# Graphical Reflections in Geometry

Sep 13, 2022

## 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

1. What transformation is shown in the below image?

Reflection along Horizontal line

1. What transformation is shown in the below image?

Reflection along Horizontal line

1. Which image shows reflection transformation?

### 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

### 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

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division  A → Addition S → Subtraction         Some examples […]