Answer :
To determine which transformations could have moved the vertex from [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex], we need to analyze the effect of each transformation on the coordinates.
For rotation transformations around the origin, the rules are as follows:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: Rotating a point [tex]\((x, y)\)[/tex] by [tex]\(90^{\circ}\)[/tex] counter-clockwise around the origin transforms it to [tex]\((-y, x)\)[/tex].
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: Rotating a point [tex]\(( x, y )\)[/tex] by [tex]\(180^{\circ}\)[/tex] around the origin transforms it to [tex]\((- x, - y)\)[/tex].
3. [tex]\(R_{0,270^{\circ}}\)[/tex]: Rotating a point [tex]\(( x, y )\)[/tex] by [tex]\(270^{\circ}\)[/tex] counter-clockwise, or [tex]\(90^{\circ}\)[/tex] clockwise around the origin transforms it to [tex]\(( y, - x )\)[/tex].
4. [tex]\(R_{0,-90^{\circ}}\)[/tex]: Rotating a point [tex]\(( x, y )\)[/tex] by [tex]\(-90^{\circ}\)[/tex] counter-clockwise, or [tex]\(270^{\circ}\)[/tex] clockwise around the origin transforms it to [tex]\(( y, - x )\)[/tex].
5. [tex]\(R_{0,-180^{\circ}}\)[/tex]: Rotating a point [tex]\(( x, y )\)[/tex] by [tex]\(-180^{\circ}\)[/tex], which is the same as [tex]\(180^{\circ}\)[/tex] clockwise rotation around the origin, transforms it to [tex]\((- x, - y)\)[/tex].
Given the initial coordinates [tex]\((0, 5)\)[/tex], let's apply these transformations and see which ones result in [tex]\((5, 0)\)[/tex]:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((-5, 0)\)[/tex]. This is not the target point.
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((0, -5)\)[/tex]. This is not the target point.
3. [tex]\(R_{0,270^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex]. This is one possible transformation.
4. [tex]\(R_{0,-90^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex]. This is another possible transformation.
5. [tex]\(R_{0,-180^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((0, -5)\)[/tex]. This is not the target point.
Therefore, the transformations that could have moved the vertex from [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex] are:
[tex]\[ R_{0,270^{\circ}} \][/tex]
[tex]\[ R_{0,-90^{\circ}} \][/tex]
For rotation transformations around the origin, the rules are as follows:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: Rotating a point [tex]\((x, y)\)[/tex] by [tex]\(90^{\circ}\)[/tex] counter-clockwise around the origin transforms it to [tex]\((-y, x)\)[/tex].
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: Rotating a point [tex]\(( x, y )\)[/tex] by [tex]\(180^{\circ}\)[/tex] around the origin transforms it to [tex]\((- x, - y)\)[/tex].
3. [tex]\(R_{0,270^{\circ}}\)[/tex]: Rotating a point [tex]\(( x, y )\)[/tex] by [tex]\(270^{\circ}\)[/tex] counter-clockwise, or [tex]\(90^{\circ}\)[/tex] clockwise around the origin transforms it to [tex]\(( y, - x )\)[/tex].
4. [tex]\(R_{0,-90^{\circ}}\)[/tex]: Rotating a point [tex]\(( x, y )\)[/tex] by [tex]\(-90^{\circ}\)[/tex] counter-clockwise, or [tex]\(270^{\circ}\)[/tex] clockwise around the origin transforms it to [tex]\(( y, - x )\)[/tex].
5. [tex]\(R_{0,-180^{\circ}}\)[/tex]: Rotating a point [tex]\(( x, y )\)[/tex] by [tex]\(-180^{\circ}\)[/tex], which is the same as [tex]\(180^{\circ}\)[/tex] clockwise rotation around the origin, transforms it to [tex]\((- x, - y)\)[/tex].
Given the initial coordinates [tex]\((0, 5)\)[/tex], let's apply these transformations and see which ones result in [tex]\((5, 0)\)[/tex]:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((-5, 0)\)[/tex]. This is not the target point.
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((0, -5)\)[/tex]. This is not the target point.
3. [tex]\(R_{0,270^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex]. This is one possible transformation.
4. [tex]\(R_{0,-90^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex]. This is another possible transformation.
5. [tex]\(R_{0,-180^{\circ}}\)[/tex]: Transforms [tex]\((0, 5)\)[/tex] to [tex]\((0, -5)\)[/tex]. This is not the target point.
Therefore, the transformations that could have moved the vertex from [tex]\((0, 5)\)[/tex] to [tex]\((5, 0)\)[/tex] are:
[tex]\[ R_{0,270^{\circ}} \][/tex]
[tex]\[ R_{0,-90^{\circ}} \][/tex]
