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A graph shows four quadrants with the x-axis and y-axis both ranging from negative 5 to positive 5 in increments of 1. Unlabeled data points marked are at left parenthesis negative 2 comma negative 1 right parenthesis, and at left parenthesis 2 comma 4 right parenthesis. A straight line joins the two lines with a positive slope.

Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.

(1 point)
Responses

5.57
5.57

6.40
6.40

9
9

41
41
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1 answer

To find the distance between the two points \((-2, -1)\) and \((2, 4)\) using the Pythagorean theorem, we first calculate the difference in the x-coordinates and the difference in the y-coordinates.

  1. Calculate the differences:

    • Difference in x-coordinates: \[ x_2 - x_1 = 2 - (-2) = 2 + 2 = 4 \]
    • Difference in y-coordinates: \[ y_2 - y_1 = 4 - (-1) = 4 + 1 = 5 \]

  2. Apply the Pythagorean theorem: The distance \(d\) between the two points can be found using the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the differences: \[ d = \sqrt{(4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41} \]

  3. Calculate \(\sqrt{41}\): Using a calculator or by estimating, \(\sqrt{41} \approx 6.4031\).

  4. Round the answer to the nearest hundredth: \[ d \approx 6.40 \]

So the length between the two points, rounded to the nearest hundredth, is 6.40.