To find the distance between two points in a Cartesian plane, we use the distance formula, which is an application of the Pythagorean theorem. The distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's apply this step-by-step to the pair of points [tex]\((5, -2)\)[/tex] and [tex]\((-8, -3)\)[/tex]:
1. Identify the coordinates:
- Point 1: [tex]\((5, -2)\)[/tex]
- Point 2: [tex]\((-8, -3)\)[/tex]
2. Calculate the differences in the coordinates:
- Difference in [tex]\(x\)[/tex]-coordinates: [tex]\((x_2 - x_1) = (-8) - 5 = -13\)[/tex]
- Difference in [tex]\(y\)[/tex]-coordinates: [tex]\((y_2 - y_1) = (-3) - (-2) = -3 + 2 = -1\)[/tex]
3. Square the differences:
- [tex]\((x_2 - x_1)^2 = (-13)^2 = 169\)[/tex]
- [tex]\((y_2 - y_1)^2 = (-1)^2 = 1\)[/tex]
4. Sum the squared differences:
- [tex]\(169 + 1 = 170\)[/tex]
5. Take the square root of the sum:
- [tex]\(\sqrt{170} \approx 13.038404810405298\)[/tex]
Thus, the distance between the points [tex]\((5, -2)\)[/tex] and [tex]\((-8, -3)\)[/tex] is approximately [tex]\(13.038\)[/tex].
For the special case mentioned in the question, the distance from a point [tex]\((x,y)\)[/tex] to the origin [tex]\((0,0)\)[/tex] can be found using the same distance formula. Specifically:
[tex]\[ d = \sqrt{x^2 + y^2} \][/tex]
This distance formula is indeed an extension of the Pythagorean theorem applied generally to any two points in a plane.
Therefore, the important points to note are:
- Distance between [tex]\((5, -2)\)[/tex] and [tex]\((-8, -3)\)[/tex] is approximately [tex]\(13.038\)[/tex].
- The distance formula [tex]\(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)[/tex] can be used for any two points in the Cartesian plane.
- For points measured from the origin, the formula simplifies to [tex]\(\sqrt{x^2 + y^2}\)[/tex].
Thank you for completing the exercises! Well done!
