To find the distance between the two points [tex]\((-3, 1)\)[/tex] and [tex]\((-2, 7)\)[/tex] on a grid, we can use the distance formula, which is derived from the Pythagorean theorem. The distance formula is given by:
[tex]\[ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here is a detailed step-by-step solution:
1. Identify the coordinates of the points:
- The first point [tex]\((x_1, y_1) = (-3, 1)\)[/tex]
- The second point [tex]\((x_2, y_2) = (-2, 7)\)[/tex]
2. Calculate the differences in the x-coordinates and y-coordinates:
- The difference in the x-coordinates ([tex]\( \Delta x \)[/tex]):
[tex]\[
\Delta x = x_2 - x_1 = -2 - (-3) = -2 + 3 = 1
\][/tex]
- The difference in the y-coordinates ([tex]\( \Delta y \)[/tex]):
[tex]\[
\Delta y = y_2 - y_1 = 7 - 1 = 6
\][/tex]
3. Substitute [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex] into the distance formula:
[tex]\[
\text{distance} = \sqrt{(1)^2 + (6)^2}
\][/tex]
4. Square the differences:
[tex]\[
1^2 = 1
\][/tex]
[tex]\[
6^2 = 36
\][/tex]
5. Add the squared differences:
[tex]\[
1 + 36 = 37
\][/tex]
6. Take the square root of the sum:
[tex]\[
\sqrt{37} \approx 6.082762530298219
\][/tex]
Thus, the distance between the points [tex]\((-3, 1)\)[/tex] and [tex]\((-2, 7)\)[/tex] is approximately [tex]\(6.082762530298219\)[/tex].
