Graphical Reflections in Geometry

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? 

Answer: 

Reflection along Horizontal line

2. What transformation is shown in the below image? 

Answer: 

Reflection along Horizontal line 

3. 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