D AY 48 – C OMPOSITION OF RIGID MOTIONS
I NTRODUCTION In the previous lessons, we took an in-depth study on the basic rigid motions, basically, rotations, reflections, translations and glide reflections, including their key properties. It is possible to perform a combination of two or more rigid motions on a given pre-image, that is, using the image of the first rigid motion as the pre- image for the next rigid motion. In this lesson, we will show that a combination of two rigid motions is also a rigid motion and when a half-turn and a reflection are combined on a given pre-image, the effect is the same as transforming the pre-image through a glide reflection.
V OCABULARY 1. Rigid motion A transformation which changes the position of a plane figure without changing the figure’s shape or size. It is also called a rigid transformation. 2. Glide reflection A transformation where the pre-image is reflected and translated parallel to the line of reflection. The order does not matter.
3. Composition of rigid motions If a certain rigid motion is applied to a plane figure and then one or more rigid motions applied to its image, the result is called a composition of rigid motions. It is also referred to as a sequence of rigid motions
We know that under any type of rigid motion: 1. Lengths of line segments on the plane figure are preserved 2. Angle measures of the plane figure are preserved 3. Collinear points remain collinear after the transformation 4. Parallel lines remain parallel after the transformation
In a nutshell, the image is exactly the same size and shape as the pre-image. Considering these key properties of rigid motion, we can show that any composition of two rigid motions is also a rigid motion. The rigid motions can be of one type or different types.
T HE COMPOSITION OF TWO RIGID MOTIONS A rigid motion preserves lengths and angle measures, implying that regardless of the number of rigid motions performed, the final image will have the same size and shape as the original pre-image. In a case of two rigid motions, lengths and angle measures will be preserved after the two rigid motions on the pre-image.
We normally use the symbol, ∘ to represent a composition of rigid motions, for example, 𝒔 𝒛−𝒃𝒚𝒋𝒕 ∘ 𝑼 𝟔,𝟓 , means a translation of 𝑦 + 5, 𝑧 + 4 followed by a reflection along the y -axis. Note that the transformations are performed from the right towards the left .
Theorem: The composition of two rigid motions is also a rigid motion. Explanation: This implies that if we perform any rigid motion on a given pre-image, then we treat the resulting image as our new pre-image and then perform the second rigid transformation, the resulting image will be the same shape and size are the original pre-image. Shape and size is preserved after the two transformations. We can use any two consecutive rigid motions on a given plane figure to show this.
Let us perform a rotation, followed by a translation of the triangle below. Note that both a translation and a rotation are rigid motions. We will compare the final image to the original pre-image afterward.
Consider Δ ABC on the grid below. 5 4 B′ C′ 3 2 1 A′ − 2 − 1 3 7 −5 −4 −3 1 2 4 6 −6 5 A B′′ −1 C′′ − 2 −3 − 4 B C A′′
The following is the sequence of the mappings. 𝚬 𝐁𝐂𝐃 → 𝚬 𝐁′𝐂′𝐃′ → 𝚬 𝐁′′𝐂′′𝐃′′ Δ ABC has been rotated through 180° about the origin 0,0 to Δ A ′ B ′ C′ . Δ A ′ B ′ C′ has been translated 5 units downwards to Δ A ′′ B′′C′′ . We want to show that Δ ABC and Δ A ′′ B′′C ′′ are congruent.
The composition of rigid motions : There are two rigid motions, a rotation followed by a reflection. Now, considering Δ ABC and Δ A ′′ B′′C ′′ ; the corresponding sides are: AB and A ′′ B ′′ ; BC and B ′′ C ′′ ; AC and A ′′ C′′ . Clearly, from the diagram above, these corresponding sides are equal and we see that AB → A ′′ B ′′ ; BC → B ′′ C ′′ and AC → A ′′ C′′
Similarly, the corresponding angles are mapped as follows: ∠𝐵 → ∠𝐵′′ , ∠𝐶 → ∠𝐶′′ and ∠𝐷 → ∠𝐷′′ . From the diagram above, these corresponding angles are also equal. Basing on the concept of triangle congruence , two triangles are congruent if all the corresponding three sides are equal and all the corresponding three angles ae equal. This show that Δ ABC ≅ Δ A ′′ B′′C ′′ . It is now evident that the composition of two rigid motions is also a rigid motion.
T HE COMPOSITION OF A HALF - TURN AND REFLECTION This is a sequence two rigid motions, a rotation of a plane figure through 180° about a given point followed by a reflection of the resulting pre-image.
Theorem: The composition of a half-turn and a reflection is a glide reflection if the center of rotation is not on the line of reflection. Note In a case where the center is on the line of reflection, the composition becomes a glide reflection where the translation involved is of zero distance, that is a mere reflection. Thus, this can be termed as a special case of the line.
Consider Δ ABC on the grid below. y 5 4 B′ C′ 3 2 1 A′ x − 2 3 4 −5 1 2 5 −4 − 1 7 −6 −3 0 6 A A′′ −1 − 2 −3 − 4 B C B′′ C′′
There are two rigid motions, a rotation of 180° about the origin (0,0) followed by a reflection along the 𝑦 − 𝑏𝑦𝑗𝑡 . We can see that the center of rotation is on the line of reflection, therefore, the composition becomes a mere reflection . The following is the sequence of the mappings. 𝚬 𝐁𝐂𝐃 → 𝚬 𝐁′𝐂′𝐃′ → 𝚬 𝐁′′𝐂′′𝐃′′
Considering Δ ABC and Δ A ′′ B′′C ′′ ; the corresponding sides are: AB and A ′′ B ′′ ; BC and B ′′ C ′′ ; AC and A ′′ C′′ . Clearly, from the diagram above, these corresponding sides are equal and we see that AB → A ′′ B ′′ ; BC → B ′′ C ′′ and AC → A ′′ C′′ . Under a reflection distance is preserved, the corresponding sides are equal.
Similarly, the corresponding angles are mapped as follows: ∠𝐵 → ∠𝐵′′ , ∠𝐶 → ∠𝐶′′ and ∠𝐷 → ∠𝐷′′ . From the diagram above, these corresponding angles are also equal. Under a reflection angle measures are preserved. It is clear that points remain on the same lines from the two triangles, for example points A and B on the side AB remain on the same line A ′′ B′′ . Basing on the concept of triangle congruence , two triangles are congruent if all the corresponding three sides are equal and all the corresponding three angles ae equal. This show that Δ ABC ≅ Δ A ′′ B′′C ′′ .
We can also note that orientation is not preserved. Considering the properties of a reflection, Δ ABC is reflected onto Δ A ′′ B′′C ′′ . We have shown that when the center is on the line of reflection, the composition becomes a reflection.
We now want to show that the composition of a half-turn and a reflection is a glide reflection if the center of rotation is not on the line of reflection. We will perform a half-turn on Δ ABC about the origin (0,0) first; then we will reflect Δ A′B′C′ over any line of reflection which does not pass through the origin, which is the center of rotation in this case, say 𝑦 = 1 . The following is the sequence of the mappings. 𝚬 𝐁𝐂𝐃 → 𝚬 𝐁′𝐂′𝐃′ → 𝚬 𝐁′′𝐂′′𝐃′′
The line reflection, 𝑦 = 1 is dotted. y 5 4 B′ C′ B′′ C′′ 3 2 1 A′′ A′ x 0 1 2 3 5 6 7 − 2 4 − 1 −4 −3 −5 −6 A −1 − 2 −3 − 4 B C
Let us consider Δ ABC and Δ A ′′ B′′C ′′ in the diagram above. If we reflect Δ ABC with the 𝑦 − 𝑏𝑦𝑗𝑡 as the line of reflection, the image of Δ ABC becomes the dotted triangle as shown below.
The dotted triangle is the image of Δ ABC . y 5 4 B′′ B′ C′ C′′ 3 2 1 A′ A′′ 0 x 1 2 − 1 3 −5 4 −4 6 − 2 5 −6 7 −3 A −1 − 2 −3 − 4 B C
If we translate the dotted triangle 2 units to the left, it coincides with Δ A ′′ B′′C ′′ . Δ ABC is mapped onto Δ A ′′ B′′C ′′ through a glide reflection. This clearly shows that the composition of a half- turn and a reflection is a glide reflection if the center of rotation is not on the line of reflection.
Example The figure below represents a composition of two rigid motions on ΔKLM . K L M Mirror line , l L′ M′ L′′ M′′ K′ K′′
(a) State the sequence of rigid motions. (b) Show how the corresponding sides on ΔKLM are mapped onto ΔK ′′ L ′′ M ′′ . (c) Show how the corresponding angles on ΔKLM are mapped onto ΔK ′′ L ′′ M ′′ . (d) Basing on (b) and (c) above, what is the relationship between ΔKLM and ΔK ′′ L ′′ M ′′ ? (e) What is your conclusion about the two rigid motions above based on (b), (c) and (d) above? (f) Which type of rigid motion maps ΔKLM onto ΔK ′′ L ′′ M ′′ in the sequence shown above?
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