Question

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.

Answer 1

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.