IDNLearn.com: Where your questions meet expert advice and community support. Discover the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
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].
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.
