Answer :

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]