To find the length of the line segment between the points \((-2, 1)\) and \((4, 3)\), we can use the distance formula, which is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For our points, let's denote \((x_1, y_1) = (-2, 1)\) and \((x_2, y_2) = (4, 3)\).
Now, calculate the differences:
\[ x_2 - x_1 = 4 - (-2) = 4 + 2 = 6 \] \[ y_2 - y_1 = 3 - 1 = 2 \]
Now we can plug these values into the distance formula:
\[ d = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} \]
So, the length of the red line segment is \(\sqrt{40}\) units.
Now let's look at the statements provided:
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The length is \(40−−√\) units; find the sum of the vertical and horizontal distances squared and then take the square root.
- This is true, as we calculated \( \sqrt{40} \).
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The length is square root of 40 units; find the sum of the vertical and horizontal distances squared and then take the square root.
- This is also true, as \(d = \sqrt{40}\).
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The length is \(260−−−√\); find the difference of the x coordinates and the difference of the y coordinates and subtract those differences.
- This is false; we do not subtract the distances to find the length.
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The length is square root of 260; find the difference of the x coordinates and the difference of the y coordinates and subtract those differences.
- This is false for the same reason as above.
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The length is 40 units; find the sum of the x and y coordinates and square root the answer.
- This is false, because the correct answer is not 40 units, but rather \(\sqrt{40}\).
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The length is 40 units; find the sum of the x and y coordinates and square root the answer.
- This is false for the same reason as above.
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The length is 130; find the difference between the x and y coordinates and square it.
- This is also false since the length is not calculated this way.
Thus, the correct statements are:
- The length is \(\sqrt{40}\) units; and
- The length is square root of 40 units; both are true.