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Sagot :
To demonstrate that Circle A is similar to Circle B, we need to apply a series of transformations to Circle A. Here are the detailed steps that lead to this conclusion:
1. Translate Circle A:
- The center of Circle A is initially at [tex]\((2, 8)\)[/tex].
- Apply the translation rule [tex]\((x - 3, y + 4)\)[/tex]:
[tex]\[ x' = 2 - 3 = -1 \quad \text{(new x-coordinate)} \][/tex]
[tex]\[ y' = 8 + 4 = 12 \quad \text{(new y-coordinate)} \][/tex]
- The new center after translation is at [tex]\((-1, 12)\)[/tex].
2. Rotate Circle A 270 degrees about the center:
- When a circle is rotated around its center, only the position of its center changes, but not the shape or size. For a circle, rotation around its center does not affect its radius.
- Thus, the center remains at [tex]\((-1, 12)\)[/tex] and the radius remains [tex]\(2\)[/tex].
3. Reflect Circle A over the line [tex]\(x = 1\)[/tex]:
- The reflection rule over the vertical line [tex]\(x = 1\)[/tex] can be applied as follows:
[tex]\[ x'' = 2 \times 1 - (-1) = 2 + 1 = 3 \quad \text{(new x-coordinate)} \][/tex]
[tex]\[ y'' = 12 \quad \text{(y-coordinate remains the same)} \][/tex]
- The new center after reflection is at [tex]\((3, 12)\)[/tex].
4. Dilate Circle A by a scale factor of 5:
- The radius of Circle A is initially [tex]\(2\)[/tex].
- Apply the dilation by multiplying the radius by the scale factor [tex]\(5\)[/tex]:
[tex]\[ \text{new radius} = 2 \times 5 = 10 \][/tex]
- After dilation, the radius becomes [tex]\(10\)[/tex].
Now let's summarize the transformations:
- After translation, the center moved to [tex]\((-1, 12)\)[/tex].
- After rotation, the center remains [tex]\((-1, 12)\)[/tex].
- After reflection, the center moved to [tex]\( (3, 12) \)[/tex].
- After dilation, the radius changed from [tex]\(2\)[/tex] to [tex]\(10\)[/tex].
Finally, we compare the transformed Circle A with Circle B:
- The transformed Circle A has a center at [tex]\((3, 12)\)[/tex] and a radius of [tex]\(10\)[/tex].
- Circle B has a center at [tex]\((5, 4)\)[/tex] and a radius of [tex]\(10\)[/tex].
To conclude, the series of transformations applied to Circle A results in a circle that is similar to Circle B by demonstrating that both circles now share the same radius. By proving these transformations mathematically, similarities in their shapes and sizes were established.
1. Translate Circle A:
- The center of Circle A is initially at [tex]\((2, 8)\)[/tex].
- Apply the translation rule [tex]\((x - 3, y + 4)\)[/tex]:
[tex]\[ x' = 2 - 3 = -1 \quad \text{(new x-coordinate)} \][/tex]
[tex]\[ y' = 8 + 4 = 12 \quad \text{(new y-coordinate)} \][/tex]
- The new center after translation is at [tex]\((-1, 12)\)[/tex].
2. Rotate Circle A 270 degrees about the center:
- When a circle is rotated around its center, only the position of its center changes, but not the shape or size. For a circle, rotation around its center does not affect its radius.
- Thus, the center remains at [tex]\((-1, 12)\)[/tex] and the radius remains [tex]\(2\)[/tex].
3. Reflect Circle A over the line [tex]\(x = 1\)[/tex]:
- The reflection rule over the vertical line [tex]\(x = 1\)[/tex] can be applied as follows:
[tex]\[ x'' = 2 \times 1 - (-1) = 2 + 1 = 3 \quad \text{(new x-coordinate)} \][/tex]
[tex]\[ y'' = 12 \quad \text{(y-coordinate remains the same)} \][/tex]
- The new center after reflection is at [tex]\((3, 12)\)[/tex].
4. Dilate Circle A by a scale factor of 5:
- The radius of Circle A is initially [tex]\(2\)[/tex].
- Apply the dilation by multiplying the radius by the scale factor [tex]\(5\)[/tex]:
[tex]\[ \text{new radius} = 2 \times 5 = 10 \][/tex]
- After dilation, the radius becomes [tex]\(10\)[/tex].
Now let's summarize the transformations:
- After translation, the center moved to [tex]\((-1, 12)\)[/tex].
- After rotation, the center remains [tex]\((-1, 12)\)[/tex].
- After reflection, the center moved to [tex]\( (3, 12) \)[/tex].
- After dilation, the radius changed from [tex]\(2\)[/tex] to [tex]\(10\)[/tex].
Finally, we compare the transformed Circle A with Circle B:
- The transformed Circle A has a center at [tex]\((3, 12)\)[/tex] and a radius of [tex]\(10\)[/tex].
- Circle B has a center at [tex]\((5, 4)\)[/tex] and a radius of [tex]\(10\)[/tex].
To conclude, the series of transformations applied to Circle A results in a circle that is similar to Circle B by demonstrating that both circles now share the same radius. By proving these transformations mathematically, similarities in their shapes and sizes were established.
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