To find the distance between two points [tex]\(A(6, 2)\)[/tex] and [tex]\(B(1, -4)\)[/tex] in a two-dimensional plane, we can use the distance formula derived from the Pythagorean theorem.
The distance [tex]\(d\)[/tex] between points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here’s a step-by-step solution:
1. Identify the coordinates:
- Point A has the coordinates [tex]\((x_1, y_1) = (6, 2)\)[/tex].
- Point B has the coordinates [tex]\((x_2, y_2) = (1, -4)\)[/tex].
2. Calculate the differences in the [tex]\(x\)[/tex]-coordinates and [tex]\(y\)[/tex]-coordinates:
- [tex]\(\Delta x = x_2 - x_1 = 1 - 6 = -5\)[/tex]
- [tex]\(\Delta y = y_2 - y_1 = -4 - 2 = -6\)[/tex]
3. Square each of these differences:
- [tex]\((\Delta x)^2 = (-5)^2 = 25\)[/tex]
- [tex]\((\Delta y)^2 = (-6)^2 = 36\)[/tex]
4. Sum these squared differences:
- [tex]\(25 + 36 = 61\)[/tex]
5. Take the square root of the sum to find the distance:
- [tex]\(d = \sqrt{61} \approx 7.81\)[/tex]
Thus, the distance between points [tex]\(A(6, 2)\)[/tex] and [tex]\(B(1, -4)\)[/tex] is approximately [tex]\(7.81\)[/tex].
Possible Errors:
- Not squaring the differences: If someone calculates [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex] correctly but forgets to square them before summing, they might sum [tex]\( \Delta x + \Delta y \)[/tex] instead.
- Not taking the square root of the sum: If someone computes [tex]\( (\Delta x)^2 + (\Delta y)^2 \)[/tex] correctly but forgets to take the square root, they would result in [tex]\( 61 \)[/tex] instead of [tex]\( \sqrt{61} \)[/tex].
Given the correct calculations:
- [tex]\(\Delta x = -5\)[/tex]
- [tex]\(\Delta y = -6\)[/tex]
- Distance: [tex]\( \sqrt{61} \approx 7.81 \)[/tex]
The correct distance between points [tex]\(A(6, 2)\)[/tex] and [tex]\(B(1, -4)\)[/tex] is approximately [tex]\(7.81\)[/tex].
Thus, if there were errors in your calculations, they might have been in forgetting to square the differences or in not taking the square root at the end.
