Let's determine the distance between the points (-2, -20) and (-2, -15).
Here's a step-by-step solution for calculating the distance between two points in a Cartesian coordinate system:
1. Identify the coordinates of the points:
- Point 1: [tex]\((x1, y1) = (-2, -20)\)[/tex]
- Point 2: [tex]\((x2, y2) = (-2, -15)\)[/tex]
2. Use the distance formula:
The distance [tex]\(d\)[/tex] between two points [tex]\((x1, y1)\)[/tex] and [tex]\((x2, y2)\)[/tex] in the Cartesian coordinate system is given by the formula:
[tex]\[
d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
\][/tex]
3. Substitute the coordinates into the formula:
[tex]\[
d = \sqrt{((-2 - (-2))^2 + (-15 - (-20))^2)}
\][/tex]
4. Simplify the expressions inside the square root:
- The difference in the x-coordinates:
[tex]\[
x2 - x1 = -2 - (-2) = -2 + 2 = 0
\][/tex]
- The difference in the y-coordinates:
[tex]\[
y2 - y1 = -15 - (-20) = -15 + 20 = 5
\][/tex]
5. Substitute these differences back into the formula:
[tex]\[
d = \sqrt{(0)^2 + (5)^2}
\][/tex]
6. Calculate the squares:
[tex]\[
d = \sqrt{0 + 25}
\][/tex]
7. Add the squared terms and take the square root:
[tex]\[
d = \sqrt{25} = 5
\][/tex]
8. State the final result:
The distance between the points [tex]\((-2, -20)\)[/tex] and [tex]\((-2, -15)\)[/tex] is [tex]\(5\)[/tex].
Thus, the coordinates for points (-2, -20), (-2, -15) and the calculated distance (5.0) are correct.
