Use the image to answer the question.

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

5.57
5.57

6.40
6.40

9
9

Answer 1

To find the length between the two points \((-2, -1)\) and \((2, 4)\), we can use the distance formula, which is derived from the Pythagorean Theorem. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting in the points \((-2, -1)\) and \((2, 4)\):

• \(x_1 = -2\)
• \(y_1 = -1\)
• \(x_2 = 2\)
• \(y_2 = 4\)

Now, calculating the differences:

\[ x_2 - x_1 = 2 - (-2) = 2 + 2 = 4 \] \[ y_2 - y_1 = 4 - (-1) = 4 + 1 = 5 \]

Now, substitute these values into the distance formula:

\[ d = \sqrt{(4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41} \]

Calculating \(\sqrt{41}\):

\[ \sqrt{41} \approx 6.403124237 \]

Rounding to the nearest hundredth:

\[ d \approx 6.40 \]

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