To find the length of the line segment, denoted as [tex]\(\overline{AB}\)[/tex], between two points [tex]\( A = (7, -1) \)[/tex] and [tex]\( B = (3, 12) \)[/tex], we use the Distance Formula. The Distance Formula is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, the coordinates of the two points are:
- Point [tex]\( A \)[/tex] has coordinates [tex]\((x_1, y_1) = (7, -1)\)[/tex]
- Point [tex]\( B \)[/tex] has coordinates [tex]\((x_2, y_2) = (3, 12)\)[/tex]
Now, substituting in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(3 - 7)^2 + (12 + 1)^2} \][/tex]
First, calculate the differences in the x-coordinates and y-coordinates:
[tex]\[ x_2 - x_1 = 3 - 7 = -4 \][/tex]
[tex]\[ y_2 - y_1 = 12 - (-1) = 12 + 1 = 13 \][/tex]
Next, square these differences:
[tex]\[ (-4)^2 = 16 \][/tex]
[tex]\[ 13^2 = 169 \][/tex]
Then, add the squared differences:
[tex]\[ 16 + 169 = 185 \][/tex]
Finally, take the square root of the sum:
[tex]\[ \sqrt{185} \approx 13.601 \][/tex]
Therefore, the length of [tex]\(\overline{AB}\)[/tex] is approximately [tex]\( 13.601 \)[/tex] units.
