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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

41
41
6:40
6:40
9
9
5.57
5.57

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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. Data points marked are point A at left parenthesis 2 comma negative 3 right parenthesis, and point B at left parenthesis 5 comma 5 right parenthesis.

1 answer

To find the distance between the two points using the Pythagorean Theorem, we can treat the distance as the hypotenuse of a right triangle, where the legs are the differences in the x-coordinates and y-coordinates.

The two points are:

  • Point A: \((-2, -1)\)
  • Point B: \((2, 4)\)

  1. Calculate the difference in the x-coordinates: \[ \Delta x = x_2 - x_1 = 2 - (-2) = 2 + 2 = 4 \]

  2. Calculate the difference in the y-coordinates: \[ \Delta y = y_2 - y_1 = 4 - (-1) = 4 + 1 = 5 \]

  3. Apply the Pythagorean theorem to find the distance \(d\): \[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41} \]

  4. Evaluate \(\sqrt{41}\): \[ \sqrt{41} \approx 6.4 \]

So the distance between the two points \((-2, -1)\) and \((2, 4)\) is approximately 6.4 when rounded to the nearest hundredth.

Now, for the second set of points:

  • Point A: \((2, -3)\)
  • Point B: \((5, 5)\)

  1. Calculate the difference in the x-coordinates: \[ \Delta x = 5 - 2 = 3 \]

  2. Calculate the difference in the y-coordinates: \[ \Delta y = 5 - (-3) = 5 + 3 = 8 \]

  3. Apply the Pythagorean theorem to find the distance \(d\): \[ d = \sqrt{(3)^2 + (8)^2} = \sqrt{9 + 64} = \sqrt{73} \]

  4. Evaluate \(\sqrt{73}\): \[ \sqrt{73} \approx 8.54 \]

So the distance between the points \((2, -3)\) and \((5, 5)\) is approximately 8.54 when rounded to the nearest hundredth.