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The image is a graph showing a straight line connecting two points, (-2, 1) and (4, 3).

The image depicts a Cartesian coordinate system, a grid with x and y axes. The axes are clearly labeled, although the numbers are somewhat small. The background is a pale, light blue. A straight, reddish-pink line is drawn connecting two points. These points are clearly marked and labeled with their coordinates: (-2, 1) and (4, 3). Each point is circled, making them visually stand out from the line. The overall mood is objective and purely informative; it's a simple, clear mathematical representation. There is no emotional content or artistic expression in the image. The atmosphere is sterile and academic, typical of a mathematical graph.

Put the steps in order to find the distance between these 2 points.

(7 points)
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Step 1
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Step 7

1 answer

To find the distance between the two points (-2, 1) and (4, 3), you can follow these steps:

Step 1: Identify the coordinates of the points.

  • The first point is (-2, 1).
  • The second point is (4, 3).

Step 2: Use the distance formula.

  • Recall the distance formula:
    \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Assign the coordinates.

  • Let \( (x_1, y_1) = (-2, 1) \) and \( (x_2, y_2) = (4, 3) \).

Step 4: Calculate \( x_2 - x_1 \).

  • Compute:
    \( x_2 - x_1 = 4 - (-2) \)
    \( = 4 + 2 = 6 \)

Step 5: Calculate \( y_2 - y_1 \).

  • Compute:
    \( y_2 - y_1 = 3 - 1 = 2 \)

Step 6: Plug into the distance formula.

  • Substitute the differences back into the formula: \[ d = \sqrt{(6)^2 + (2)^2} \]

Step 7: Calculate the distance.

  • Simplify the equation:
    \( d = \sqrt{36 + 4} = \sqrt{40} \)
    \( = \sqrt{4 \cdot 10} = 2\sqrt{10} \)

Summary of Steps:

  1. Identify the coordinates of the points.
  2. Use the distance formula.
  3. Assign the coordinates.
  4. Calculate \( x_2 - x_1 \).
  5. Calculate \( y_2 - y_1 \).
  6. Plug into the distance formula.
  7. Calculate the distance.

This structured approach leads you logically through the process of finding the distance between two points in a Cartesian coordinate system.