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Find the length of the red line segment in the coordinate plane.

A graph, likely part of a math problem, is presented on a gridded coordinate plane. 

A straight red line is drawn on the graph, passing through two marked points. The coordinates of the points are labeled on the graph:

(-2, 1)
(4, 3)

The coordinate axes are clearly labeled (x and y). The graph appears to be showing a linear relationship between the x and y values.  The gridlines are standard for graph paper. The line starts below the x-axis passing right through the point (-2, 1.) and continues upwards to the right to pass through the point (4, 3.)

Which of the following statements are true:

(1 point)
Responses

The length is 130; find the difference between the x and y coordinates and square it.
The length is 130; find the difference between the x and y coordinates and square it.

The length is 40 units; find the sum of the x and y coordinates and square root the answer.
The length is 40 units; find the sum of the x and y coordinates and square root the answer.

The length is 40−−√ units; find the sum of the vertical and horizontal distances squared and then take the square root.
The length is square root of 40 units; find the sum of the vertical and horizontal distances squared and then take the square root.

The length is 260−−−√; find the difference of the x coordinates and the difference of the y coordinates and subtract those differences.
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.

1 answer

To find the length of the red line segment between the points \((-2, 1)\) and \((4, 3)\), we can use the distance formula, which is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, \((x_1, y_1) = (-2, 1)\) and \((x_2, y_2) = (4, 3)\). Let's calculate the differences:

  • \(x_2 - x_1 = 4 - (-2) = 4 + 2 = 6\)
  • \(y_2 - y_1 = 3 - 1 = 2\)

Now, we can substitute these values into the distance formula:

\[ d = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} \]

Thus, the length of the red line segment is \(\sqrt{40}\) units.

Looking at the provided statements, the correct one is:

  • The length is \(\sqrt{40}\) units; find the sum of the vertical and horizontal distances squared and then take the square root.

This matches our calculation. The other statements are not correct based on our findings.