Compositions of Transformations

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 2x^o, where x^o 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.

2. Find the angle of rotation that maps A onto A”.

3. Find the angle of rotation that maps A onto A”.

4. Find the angle of rotation which maps A onto A”. Given x  = 66 .

5. Find the angle between A and A”.

6. Describe the type of composition of transformation.

7. Find the distance between the parallel lines if the distance ABC between A”B”C” is 1200 mm.

8. Find the distance between ABC to A”B”C” , the distance between the parallel lines k and m is 500 m.

9. 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

10. 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.

Concept Map

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