Question

Use the image to answer the 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 41415.575.576:406:409

Answer 1

To find the length between the two points \((-2, -1)\) and \((2, 4)\) using the Pythagorean Theorem, we first need to determine the horizontal and vertical distances between the two points.

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

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

3. **Use the Pythagorean Theorem**:
The length \(d\) between the two points can be found using the theorem:
\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]
Substituting the values we calculated:
\[
d = \sqrt{(4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41}
\]

4. **Calculate \(\sqrt{41}\)**:
\[
\sqrt{41} \approx 6.403124237
\]
Rounding this to the nearest hundredth gives \(6.40\).

Thus, the length between the two points is approximately **6.40**.