#### Need Help?

Get in touch with us

# Compositions of Transformations

Sep 13, 2022

## Key Concepts

• Understand two reflections
• Understand reflections in parallel lines theorem
• Understand reflections in intersecting lines theorem
• Solve reflections in parallel lines
• Solve reflections in intersecting lines

### Two reflections

Compositions of two reflections result in either a translation or a rotation.

Let us understand more about the two reflections with the help of the following theorems.

1. Reflections in parallel lines theorem
2. Reflections in intersecting lines theorem

### Reflections in parallel lines theorem

#### Theorem 9.5 Reflections in Parallel Lines Theorem

If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation.

If P” is the image of P, then

1. PP” -is perpendicular to k and m, and
2. PP”= 2d, where d is the distance between k and m.

### Reflections in intersecting lines theorem

#### Theorem 9.6 Reflections in Intersecting Lines Theorem

If lines k and m intersect at point P, then a reflection in k

Followed by a reflection in m is the same as a rotation about point p.

The angle of rotation is 2xo, where xo is the measure of the acute or right angle formed by k and m.

Note:

• A reflection followed by a reflection in parallel lines results in translation.
• A reflection followed by a reflection in intersecting lines results in rotation.

Examples:

1. In the diagram, a reflection in line k maps GH−to G′H’−. A reflection in line m maps G′H’− to G”H’−’. Also, HB = 9 and DH” = 4

Name any segments congruent to each segment: HG HB , and GA.

Does AC = BD? Explain.

What is the length of GG” ?

Solution:

1. HG- ≅≅ H’G’, and HG ≅≅ H”G”. HB ≅≅ H′B. GA≅≅ G’A
2. Yes, C = BD because GG”and HH” are perpendicular to both k and m, so BD and AC opposite sides of a rectangle.
3. By the properties of reflections, H’B = 9 and H’D = 4. Theorem 9.5 implies that GG” = HH” = 2. BD, so the length of GG” is 2(9 + 4), or 26 units.

2. In the diagram, the figure is reflected in line k. The image is then reflected in line m. Describe a single transformation that maps F to F”.

Solution:

The measure of the acute angle formed between lines k and m is 70 °.

So, by Theorem 9.6, a single transformation that maps F to F” is a 140° rotation about point P.

You can check that this is correct by tracing lines k and m and point F, then rotating the point 140°

## Exercise

1. In the diagram, the preimage is reflected in line k, then in line m. Describe a single transformation that maps the blue figure onto the green figure.
1. Find the angle of rotation that maps A onto A”.
1. Find the angle of rotation that maps A onto A”.
1. Find the angle of rotation which maps A onto A”. Given x  = 66 .
1. Find the angle between A and A”.
1. Describe the type of composition of transformation.
1. Find the distance between the parallel lines if the distance ABC between A”B”C” is 1200 mm.
1. Find the distance between ABC to A”B”C” , the distance between the parallel lines k and m is 500 m.
1. The vertices of PQR are P(1, 4), Q(3, -2), and R(7, 1). Use matrix operations to find the image matrix that represents the composition of the given transformations. Then graph PQR and its image.

Translation: (x, y) → (x, y + 5)

Reflection: In the y-axis

1. The vertices of PQR are P(1, 4), Q(3, -2), and R(7, 1). Use matrix operations to find the image matrix that represents the composition of the given transformations. Then graph PQR and its image.

Reflection: In the x-axis

Translation: (x, y) → (x-9, y – 4)

### Summary

• Compositions of two reflections result in either a translation or a rotation.
• A reflection followed by a reflection in parallel lines results in translation.
• A reflection followed by a reflection in intersecting lines results in rotation.

### What we have learned

• Understand two reflections
• Understand reflections in parallel lines theorem
• Understand reflections in intersecting lines theorem
• Solve reflections in parallel lines
• Solve reflections in intersecting lines

#### 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 […]