To find the distance between the points [tex]\((-9, -11)\)[/tex] and [tex]\((13, 3)\)[/tex], you can use the distance formula, which is derived from the Pythagorean theorem. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a coordinate plane is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, the coordinates of the two points are:
- Point 1: [tex]\((-9, -11)\)[/tex], where [tex]\(x_1 = -9\)[/tex] and [tex]\(y_1 = -11\)[/tex]
- Point 2: [tex]\((13, 3)\)[/tex], where [tex]\(x_2 = 13\)[/tex] and [tex]\(y_2 = 3\)[/tex]
Now, substituting these values into the distance formula:
1. Calculate the difference in the x-coordinates:
[tex]\[ x_2 - x_1 = 13 - (-9) = 13 + 9 = 22 \][/tex]
2. Calculate the difference in the y-coordinates:
[tex]\[ y_2 - y_1 = 3 - (-11) = 3 + 11 = 14 \][/tex]
3. Square the differences:
[tex]\[ (x_2 - x_1)^2 = 22^2 = 484 \][/tex]
[tex]\[ (y_2 - y_1)^2 = 14^2 = 196 \][/tex]
4. Add these squared differences:
[tex]\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = 484 + 196 = 680 \][/tex]
5. Take the square root of the sum to find the distance:
[tex]\[ \text{Distance} = \sqrt{680} \approx 26.076809620810597 \][/tex]
Therefore, the distance between the points [tex]\((-9, -11)\)[/tex] and [tex]\((13, 3)\)[/tex] is approximately [tex]\(26.077\)[/tex]. This can be visualized on a graph by plotting the points and using the distance formula to determine the distance between them.
