Isosceles triangles are symmetric, so when we look for the different isometries, translations as well as reflections will make the same congruent image. We can look at four possible solutions.

Basic Rigid Motions

The three basic rigid motion isometries are reflections, rotations, and translations. The first two examples show how we can map directly from the preimage to the image.

Isometry I

Let's label the top triangle $A B C$ and call it the preimage. We can arrange the bottom triangle, call it $A^{'} B^{'} C^{'}$ according to how the isometry works.

When we use this composition of $A^{'} B^{'} C^{'}$ we can look at the coordinate transformation.

Points	Coordinates
$A$ $\to$ $A^{'}$	$(3, 3) \to (3, - 3)$
$B$ $\to$ $B^{'}$	$(- 3, 5) \to (- 3, - 5)$
$C$ $\to$ $C^{'}$	$(- 3, 1) \to (- 3, - 1)$
$A B C$ $\to$ $A^{'} B^{'} C^{'}$	$(x, y)$ $\to$ $R_{x - axis}$

This transformation reflects the triangle across the $x - axis .$ Our first isometry for this transformation could be $R_{x - axis} (△ A B C) .$

Isometry II

Let's look at this in another way.

When we use this composition of $A^{'} B^{'} C^{'}$ we can look at the coordinate transformation.

Points	Coordinates
$A$ $\to$ $A^{'}$	$(3, 3) \to (3, - 3)$
$B$ $\to$ $B^{'}$	$(- 3, 5) \to (- 3, - 1)$
$C$ $\to$ $C^{'}$	$(- 3, 1) \to (- 3, - 5)$
$A B C$ $\to$ $A^{'} B^{'} C^{'}$	$(x, y)$ $\to$ $T_{< 0, - 6 >}$

This transformation shows a translation down $6$ units. Our second isometry is $T_{< 0, - 6 >} (△ A B C) .$ There are two one-step isometries.

Other Isometries

After seeing a translation and a reflection isometry, we can use our third rigid motion, rotation, in combination with the other two. The next two isometries are possible solutions, other combinations may work as well.

Isometry III

Let's get creative and look at a sequence of rigid motions to get from the top triangle to the bottom one.

Let's look at where the points landed in each transformation.

Points	Coordinates
$A$ $\to$ $A^{'}$ $\to$ $A^{''}$	$(3, 3) \to (- 9, - 3) \to (3, - 3)$
$B$ $\to$ $B^{'}$ $\to$ $B^{''}$	$(- 3, 5) \to (- 3, - 5) \to (- 3, - 5)$
$C$ $\to$ $C^{'}$ $\to$ $C^{''}$	$(- 3, 1) \to (- 3, - 1) \to (- 3, - 1)$
$A B C$ $\to$ $A^{'} B^{'} C^{'}$ $\to$ $A^{''} B^{''} C^{''}$	$(x, y)$ $\to$ $r_{(18 0^{\circ}, (- 3, 0))}$ $\to$ $R_{x = - 3}$

This rigid motion rotates $18 0^{\circ}$ around the point $(- 3, 0)$ then reflects across the line $x = - 3 .$ Our isometry for these motions is $(r_{(18 0^{\circ}, (- 3, 0))} \circ R_{x = - 3}) (△ A B C) .$

Isometry IV

Let's look at another sequence of rigid motions to get from the top triangle to the bottom one.

Let's look at where the points landed in each transformation.

Points	Coordinates
$A$ $\to$ $A^{'}$ $\to$ $A^{''}$	$(3, 3) \to (3, 3) \to (3, - 3)$
$B$ $\to$ $B^{'}$ $\to$ $B^{''}$	$(- 3, 5) \to (9, 5) \to (- 3, - 5)$
$C$ $\to$ $C^{'}$ $\to$ $C^{''}$	$(- 3, 1) \to (9, 1) \to (- 3, - 1)$
$A B C$ $\to$ $A^{'} B^{'} C^{'}$ $\to$ $A^{''} B^{''} C^{''}$	$(x, y)$ $\to$ $R_{x = 3}$ $\to$ $r_{(18 0^{\circ}, (3, 0))}$

This rigid motion reflects the triangle across the line $x = 3,$ then rotates $18 0^{\circ}$ around the point $(3, 0) .$ Our isometry for these motions is $(R_{x = 3} \circ r_{(18 0^{\circ}, (3, 0))}) (△ A B C) .$

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