To calculate the length between the points [tex]\((1, 6)\)[/tex] and [tex]\((7, 14)\)[/tex] using the distance formula, follow these steps:
1. Identify the coordinates of the points:
- The first point [tex]\( (x_1, y_1) = (1, 6) \)[/tex].
- The second point [tex]\( (x_2, y_2) = (7, 14) \)[/tex].
2. Calculate the differences in the [tex]\(x\)[/tex]-coordinates and [tex]\(y\)[/tex]-coordinates:
- Difference in [tex]\(x\)[/tex]-coordinates:
[tex]\[
\Delta x = x_2 - x_1 = 7 - 1 = 6
\][/tex]
- Difference in [tex]\(y\)[/tex]-coordinates:
[tex]\[
\Delta y = y_2 - y_1 = 14 - 6 = 8
\][/tex]
3. Apply the distance formula:
The distance formula to find the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Distance} = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\][/tex]
Substitute [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex] into the formula:
[tex]\[
\text{Distance} = \sqrt{(6)^2 + (8)^2}
\][/tex]
4. Calculate the squares and sum them:
[tex]\[
(6)^2 = 36
\][/tex]
[tex]\[
(8)^2 = 64
\][/tex]
[tex]\[
36 + 64 = 100
\][/tex]
5. Find the square root of the sum:
[tex]\[
\sqrt{100} = 10
\][/tex]
So, the length between the points [tex]\((1, 6)\)[/tex] and [tex]\((7, 14)\)[/tex] is [tex]\(10\)[/tex].
Thus, the differences in the coordinates and the distance are:
- Difference in [tex]\(x\)[/tex]-coordinates: [tex]\(6\)[/tex]
- Difference in [tex]\(y\)[/tex]-coordinates: [tex]\(8\)[/tex]
- Distance: [tex]\(10.0\)[/tex]
