for exercises 1 - 5, use the diagram.
1. what composition of two rigid motions maps △abc to △abc?
for exercises 2 - 5, find the coordinates of p under each transformation. suppose the equation of line m is y = 2 and the equation of line n is x=-1.
2. t_(-2,0)∘r_m
3. t_(0, - 5)∘r_m
4. t_(0,2)∘r_y - axis
5. t_(3,0)∘r_x - axis
for exercises 6 - 12, write a rigid motion that produces each image.
6. △abc→△def
7. △abc→△ghj
8. △abc→△klm
9. △abc→△npq
10. △abc→△rst
11. △def→△ghj
12. △ghj→△klm
13. understand define the term glide reflection.
14. apply the series of footprints can be described as a series of glide reflections. the composition of two identical glide reflections, for example, from the first step to the third, is equivalent to what rigid motion?

Question

Accepted Answer

1.
# Explanation:
## Step1: Recall identity motion
The identity motion (a translation by the zero - vector or a rotation by 0 degrees or a reflection over the same line twice etc.) maps a figure onto itself. For example, a translation $T_{(0,0)}$ followed by a rotation $R_{O,0}$ (rotation about a point $O$ by 0 degrees) will map $\triangle ABC$ to $\triangle ABC$. Another example is a reflection $r_l$ over a line $l$ followed by the same reflection $r_l$ again. The composition of two identity - like rigid motions will map the triangle onto itself.
# Answer:
A translation by the zero - vector followed by a rotation of 0 degrees (or other combinations of identity - like rigid motions such as two identical reflections)

2. Let the coordinates of point $P$ be $(x,y)$.
# Explanation:
## Step1: First perform reflection $r_m$
The equation of line $m$ is $y = 2$. The rule for reflecting a point $(x,y)$ over the line $y = k$ is $(x,2k - y)$. So, $r_m(x,y)=(x,4 - y)$
## Step2: Then perform translation $T_{(- 2,0)}$
The rule for translation $T_{(a,b)}$ is $(x,y)\to(x + a,y + b)$. So, $T_{(-2,0)}(x,4 - y)=(x-2,4 - y)$
# Answer:
$(x - 2,4 - y)$

3. Let the coordinates of point $P$ be $(x,y)$.
# Explanation:
## Step1: First perform reflection $r_m$
The equation of line $m$ is $y = 2$. The rule for reflecting a point $(x,y)$ over the line $y=k$ is $(x,2k - y)$. So, $r_m(x,y)=(x,4 - y)$
## Step2: Then perform translation $T_{(0,-5)}$
The rule for translation $T_{(a,b)}$ is $(x,y)\to(x + a,y + b)$. So, $T_{(0,-5)}(x,4 - y)=(x,4 - y-5)=(x,-y - 1)$
# Answer:
$(x,-y - 1)$

4. Let the coordinates of point $P$ be $(x,y)$.
# Explanation:
## Step1: First perform reflection $r_{y - axis}$
The rule for reflecting a point $(x,y)$ over the $y$ - axis is $(-x,y)$
## Step2: Then perform translation $T_{(0,2)}$
The rule for translation $T_{(a,b)}$ is $(x,y)\to(x + a,y + b)$. So, $T_{(0,2)}(-x,y)=(-x,y + 2)$
# Answer:
$(-x,y + 2)$

5. Let the coordinates of point $P$ be $(x,y)$.
# Explanation:
## Step1: First perform reflection $r_{x - axis}$
The rule for reflecting a point $(x,y)$ over the $x$ - axis is $(x,-y)$
## Step2: Then perform translation $T_{(3,0)}$
The rule for translation $T_{(a,b)}$ is $(x,y)\to(x + a,y + b)$. So, $T_{(3,0)}(x,-y)=(x + 3,-y)$
# Answer:
$(x + 3,-y)$

6 - 12: Without the actual diagrams showing the positions of the triangles, we can't give specific rigid - motion descriptions. In general, a rigid motion can be a translation ( $T_{(a,b)}$: $(x,y)\to(x + a,y + b)$), a rotation ($R_{O,\theta}$: rotation about a point $O$ by an angle $\theta$), or a reflection ($r_l$: reflection over a line $l$). For example, if $\triangle ABC$ and $\triangle DEF$ have the same orientation and $\triangle DEF$ is obtained by moving $\triangle ABC$ $a$ units to the right and $b$ units up, the rigid motion is $T_{(a,b)}$. If they have different orientations, it could be a rotation or a reflection followed by a translation.

13.
# Brief Explanations:
A glide reflection is a composition of a translation and a reflection. The translation and the reflection are performed in such a way that the line of reflection is parallel to the direction of the translation.
# Answer:
A glide reflection is a composition of a translation and a reflection where the line of reflection is parallel to the direction of the translation.

14.
# Explanation:
## Step1: Recall the properties of glide reflections
Let the glide reflection be a translation $T_{(a,b)}$ followed by a reflection $r_l$ where $l$ is parallel to the direction of the translation. If we perform the same glide reflection twice, the two translations will combine to form a new translation, and the two reflections over the same line will cancel each other out (since reflecting a point over a line twice returns the point to its original position in terms of reflection).
## Step2: Determine the equivalent rigid motion
The equivalent rigid motion is a translation.
# Answer:
A translation

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QUESTION IMAGE

Question

Explanation:

Step1: Recall identity motion

Step1: First perform reflection \(r_m\)

Step2: Then perform translation \(T_{(- 2,0)}\)

Step1: First perform reflection \(r_m\)

Step2: Then perform translation \(T_{(0,-5)}\)

Answer: