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Sagot :
To determine which statement correctly describes the preimage [tex]\(\overline{E F}\)[/tex] when [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex] is dilated by a scale factor of [tex]\(\frac{1}{2}\)[/tex] from the origin, follow these steps:
1. Identify the coordinates of [tex]\(E^{\prime}\)[/tex] and [tex]\(F^{\prime}\)[/tex]:
- [tex]\(E^{\prime} = (1, 0)\)[/tex]
- [tex]\(F^{\prime} = (1, 3)\)[/tex]
2. Apply the dilation with scale factor [tex]\(\frac{1}{2}\)[/tex] to these coordinates:
To calculate the coordinates of the preimage points [tex]\(E\)[/tex] and [tex]\(F\)[/tex], multiply the coordinates of [tex]\(E^{\prime}\)[/tex] and [tex]\(F^{\prime}\)[/tex] by the scale factor [tex]\(\frac{1}{2}\)[/tex]:
- For point [tex]\(E^{\prime}(1, 0)\)[/tex]:
[tex]\[ E = \left(\frac{1}{2} \times 1, \frac{1}{2} \times 0\right) = \left(0.5, 0\right) \][/tex]
- For point [tex]\(F^{\prime}(1, 3)\)[/tex]:
[tex]\[ F = \left(\frac{1}{2} \times 1, \frac{1}{2} \times 3\right) = \left(0.5, 1.5\right) \][/tex]
3. Verify the statements given:
- Statement 1: [tex]\(\overline{E F}\)[/tex] is located at [tex]\(E(0.5, 0)\)[/tex] and [tex]\(F(0.5, 1.5)\)[/tex] and is half the size of [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- This statement is correct because the coordinates match the calculated preimage coordinates, and the dilation factor ensures that [tex]\(\overline{E F}\)[/tex] is indeed half the size of [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- Statement 2: [tex]\(\overline{E F}\)[/tex] is located at [tex]\(E(1, 0)\)[/tex] and [tex]\(F(1, 3)\)[/tex] and is the same size as [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- This statement is incorrect because the coordinates of [tex]\(E\)[/tex] and [tex]\(F\)[/tex] are not transformed.
- Statement 3: [tex]\(\overline{E F}\)[/tex] is located at [tex]\(E(2, 0)\)[/tex] and [tex]\(F(2, 6)\)[/tex] and is twice the size of [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- This statement is incorrect. A scale factor of [tex]\(\frac{1}{2}\)[/tex] would not produce coordinates that double.
- Statement 4: [tex]\(\overline{E F}\)[/tex] is located at [tex]\(E(3, 0)\)[/tex] and [tex]\(F(3, 9)\)[/tex] and is three times the size of [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- This statement is incorrect because the coordinates do not reflect the correct transformation and the size factor is wrong.
4. Conclusion:
The correct statement is:
[tex]\[ \overline{E F} \text{ is located at } E(0.5,0) \text{ and } F(0.5,1.5) \text{ and is half the size of } \overline{E^{\prime} F^{\prime}}. \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{1} \][/tex]
1. Identify the coordinates of [tex]\(E^{\prime}\)[/tex] and [tex]\(F^{\prime}\)[/tex]:
- [tex]\(E^{\prime} = (1, 0)\)[/tex]
- [tex]\(F^{\prime} = (1, 3)\)[/tex]
2. Apply the dilation with scale factor [tex]\(\frac{1}{2}\)[/tex] to these coordinates:
To calculate the coordinates of the preimage points [tex]\(E\)[/tex] and [tex]\(F\)[/tex], multiply the coordinates of [tex]\(E^{\prime}\)[/tex] and [tex]\(F^{\prime}\)[/tex] by the scale factor [tex]\(\frac{1}{2}\)[/tex]:
- For point [tex]\(E^{\prime}(1, 0)\)[/tex]:
[tex]\[ E = \left(\frac{1}{2} \times 1, \frac{1}{2} \times 0\right) = \left(0.5, 0\right) \][/tex]
- For point [tex]\(F^{\prime}(1, 3)\)[/tex]:
[tex]\[ F = \left(\frac{1}{2} \times 1, \frac{1}{2} \times 3\right) = \left(0.5, 1.5\right) \][/tex]
3. Verify the statements given:
- Statement 1: [tex]\(\overline{E F}\)[/tex] is located at [tex]\(E(0.5, 0)\)[/tex] and [tex]\(F(0.5, 1.5)\)[/tex] and is half the size of [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- This statement is correct because the coordinates match the calculated preimage coordinates, and the dilation factor ensures that [tex]\(\overline{E F}\)[/tex] is indeed half the size of [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- Statement 2: [tex]\(\overline{E F}\)[/tex] is located at [tex]\(E(1, 0)\)[/tex] and [tex]\(F(1, 3)\)[/tex] and is the same size as [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- This statement is incorrect because the coordinates of [tex]\(E\)[/tex] and [tex]\(F\)[/tex] are not transformed.
- Statement 3: [tex]\(\overline{E F}\)[/tex] is located at [tex]\(E(2, 0)\)[/tex] and [tex]\(F(2, 6)\)[/tex] and is twice the size of [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- This statement is incorrect. A scale factor of [tex]\(\frac{1}{2}\)[/tex] would not produce coordinates that double.
- Statement 4: [tex]\(\overline{E F}\)[/tex] is located at [tex]\(E(3, 0)\)[/tex] and [tex]\(F(3, 9)\)[/tex] and is three times the size of [tex]\(\overline{E^{\prime} F^{\prime}}\)[/tex].
- This statement is incorrect because the coordinates do not reflect the correct transformation and the size factor is wrong.
4. Conclusion:
The correct statement is:
[tex]\[ \overline{E F} \text{ is located at } E(0.5,0) \text{ and } F(0.5,1.5) \text{ and is half the size of } \overline{E^{\prime} F^{\prime}}. \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{1} \][/tex]
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