Answer :
To determine which transformations can be used to map a triangle with vertices [tex]\( A(2,2) \)[/tex], [tex]\( B(4,1) \)[/tex], and [tex]\( C(4,5) \)[/tex] to [tex]\( A^{\prime}(-2,-2) \)[/tex], [tex]\( B^{\prime}(-1,-4) \)[/tex], and [tex]\( C^{\prime}(-5,-4) \)[/tex], we will analyze each given transformation option.
1. 180° Rotation about the origin:
- When a point [tex]\((x, y)\)[/tex] is rotated 180° about the origin, its new coordinates become [tex]\((-x, -y)\)[/tex].
- Applying this transformation:
- [tex]\(A(2,2) \)[/tex] becomes [tex]\( (-2,-2)\)[/tex],
- [tex]\(B(4,1) \)[/tex] becomes [tex]\( (-4,-1) \)[/tex] (not [tex]\((-1,-4)\)[/tex]),
- [tex]\(C(4,5) \)[/tex] becomes [tex]\( (-4,-5) \)[/tex] (not [tex]\((-5,-4)\)[/tex]).
- Thus, [tex]\(180^{\circ}\)[/tex] rotation about the origin will not map the given triangle [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle A'B'C' \)[/tex].
2. 90° Counterclockwise Rotation about the origin and a Translation down 4 units:
- When a point [tex]\((x, y) \)[/tex] is rotated 90° counterclockwise about the origin, its new coordinates become [tex]\((-y, x)\)[/tex].
- Then translating (down 4 units) [tex]\((-y, x)\)[/tex] to [tex]\((-y, x-4)\)[/tex].
- Applying this transformation:
- [tex]\(A(2,2)\)[/tex] becomes [tex]\( (-2, 2) \)[/tex] and then [tex]\((-2, 2-4)\)[/tex] which is [tex]\((-2,-2)\)[/tex],
- [tex]\(B(4,1)\)[/tex] becomes [tex]\(-1, 4)\)[/tex] and then [tex]\((-1, 4-4)\)[/tex] which is [tex]\((-1,0)\)[/tex] (not [tex]\((-1,-4)\)[/tex]),
- [tex]\(C(4,5)\)[/tex] becomes [tex]\((-5, 4)\)[/tex] and then [tex]\((-5, 4-4)\)[/tex] which is [tex]\((-5,0)\)[/tex] (not [tex]\((-5,-4)\)[/tex]).
- Hence, this transformation will not map the given triangle [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle A'B'C' \)[/tex].
3. 90° Clockwise Rotation about the origin and a Reflection over the [tex]\( y \)[/tex]-axis:
- When a point [tex]\((x, y) \)[/tex] is rotated 90° clockwise about the origin, its new coordinates become [tex]\((y, -x)\)[/tex].
- Reflecting it over the [tex]\( y\)[/tex]-axis transforms [tex]\((y, -x) \)[/tex] to [tex]\((-y, -x) \)[/tex].
- Applying this transformation:
- [tex]\(A(2,2)\)[/tex] becomes [tex]\((2, -2)\)[/tex] and then becomes [tex]\((-2,-2)\)[/tex],
- [tex]\(B(4,1)\)[/tex] becomes [tex]\( (1, -4)\)[/tex] and then becomes [tex]\((-1,-4)\)[/tex],
- [tex]\(C(4,5)\)[/tex] becomes [tex]\((5, -4)\)[/tex] and then becomes [tex]\((-5,-4)\)[/tex].
- Since [tex]\(\triangle ABC\)[/tex] effectively matches [tex]\(\triangle A'B'C'\)[/tex] after this transformation. This set of operations works.
4. Reflection over the [tex]\( y \)[/tex]-axis and then a 90° Clockwise Rotation about the origin:
- Reflecting a point [tex]\((x, y)\)[/tex] over the [tex]\( y\)[/tex]-axis gives [tex]\((-x, y)\)[/tex].
- Then rotating it 90° clockwise about the origin transforms it from [tex]\((-x, y)\)[/tex] to [tex]\((y, x)\)[/tex].
- Applying this transformation:
- [tex]\(A(2,2)\)[/tex] becomes [tex]\((-2,2)\)[/tex] and then becomes [tex]\((2, -2)\)[/tex] (not [tex]\((-2,-2)\)[/tex]),
- [tex]\(B(4,1)\)[/tex] becomes [tex]\((-4,1)\)[/tex] and then becomes [tex]\((1, -4)\)[/tex] (not [tex]\((-1,-4)\)[/tex]),
- [tex]\(C(4,5)\)[/tex] becomes [tex]\((-4,5)\)[/tex] and then becomes [tex]\((5, -4)\)[/tex] (not [tex]\((5,-4)\)[/tex]).
- This transformation does not map the given triangle [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle A'B'C' \)[/tex].
Given the transformations and their effects, the third option [tex]\("90° clockwise rotation about the origin and a reflection over the y-axis"\)[/tex] successfully maps the triangle [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle A'B'C' \)[/tex].
1. 180° Rotation about the origin:
- When a point [tex]\((x, y)\)[/tex] is rotated 180° about the origin, its new coordinates become [tex]\((-x, -y)\)[/tex].
- Applying this transformation:
- [tex]\(A(2,2) \)[/tex] becomes [tex]\( (-2,-2)\)[/tex],
- [tex]\(B(4,1) \)[/tex] becomes [tex]\( (-4,-1) \)[/tex] (not [tex]\((-1,-4)\)[/tex]),
- [tex]\(C(4,5) \)[/tex] becomes [tex]\( (-4,-5) \)[/tex] (not [tex]\((-5,-4)\)[/tex]).
- Thus, [tex]\(180^{\circ}\)[/tex] rotation about the origin will not map the given triangle [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle A'B'C' \)[/tex].
2. 90° Counterclockwise Rotation about the origin and a Translation down 4 units:
- When a point [tex]\((x, y) \)[/tex] is rotated 90° counterclockwise about the origin, its new coordinates become [tex]\((-y, x)\)[/tex].
- Then translating (down 4 units) [tex]\((-y, x)\)[/tex] to [tex]\((-y, x-4)\)[/tex].
- Applying this transformation:
- [tex]\(A(2,2)\)[/tex] becomes [tex]\( (-2, 2) \)[/tex] and then [tex]\((-2, 2-4)\)[/tex] which is [tex]\((-2,-2)\)[/tex],
- [tex]\(B(4,1)\)[/tex] becomes [tex]\(-1, 4)\)[/tex] and then [tex]\((-1, 4-4)\)[/tex] which is [tex]\((-1,0)\)[/tex] (not [tex]\((-1,-4)\)[/tex]),
- [tex]\(C(4,5)\)[/tex] becomes [tex]\((-5, 4)\)[/tex] and then [tex]\((-5, 4-4)\)[/tex] which is [tex]\((-5,0)\)[/tex] (not [tex]\((-5,-4)\)[/tex]).
- Hence, this transformation will not map the given triangle [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle A'B'C' \)[/tex].
3. 90° Clockwise Rotation about the origin and a Reflection over the [tex]\( y \)[/tex]-axis:
- When a point [tex]\((x, y) \)[/tex] is rotated 90° clockwise about the origin, its new coordinates become [tex]\((y, -x)\)[/tex].
- Reflecting it over the [tex]\( y\)[/tex]-axis transforms [tex]\((y, -x) \)[/tex] to [tex]\((-y, -x) \)[/tex].
- Applying this transformation:
- [tex]\(A(2,2)\)[/tex] becomes [tex]\((2, -2)\)[/tex] and then becomes [tex]\((-2,-2)\)[/tex],
- [tex]\(B(4,1)\)[/tex] becomes [tex]\( (1, -4)\)[/tex] and then becomes [tex]\((-1,-4)\)[/tex],
- [tex]\(C(4,5)\)[/tex] becomes [tex]\((5, -4)\)[/tex] and then becomes [tex]\((-5,-4)\)[/tex].
- Since [tex]\(\triangle ABC\)[/tex] effectively matches [tex]\(\triangle A'B'C'\)[/tex] after this transformation. This set of operations works.
4. Reflection over the [tex]\( y \)[/tex]-axis and then a 90° Clockwise Rotation about the origin:
- Reflecting a point [tex]\((x, y)\)[/tex] over the [tex]\( y\)[/tex]-axis gives [tex]\((-x, y)\)[/tex].
- Then rotating it 90° clockwise about the origin transforms it from [tex]\((-x, y)\)[/tex] to [tex]\((y, x)\)[/tex].
- Applying this transformation:
- [tex]\(A(2,2)\)[/tex] becomes [tex]\((-2,2)\)[/tex] and then becomes [tex]\((2, -2)\)[/tex] (not [tex]\((-2,-2)\)[/tex]),
- [tex]\(B(4,1)\)[/tex] becomes [tex]\((-4,1)\)[/tex] and then becomes [tex]\((1, -4)\)[/tex] (not [tex]\((-1,-4)\)[/tex]),
- [tex]\(C(4,5)\)[/tex] becomes [tex]\((-4,5)\)[/tex] and then becomes [tex]\((5, -4)\)[/tex] (not [tex]\((5,-4)\)[/tex]).
- This transformation does not map the given triangle [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle A'B'C' \)[/tex].
Given the transformations and their effects, the third option [tex]\("90° clockwise rotation about the origin and a reflection over the y-axis"\)[/tex] successfully maps the triangle [tex]\( \triangle ABC \)[/tex] to [tex]\( \triangle A'B'C' \)[/tex].
