Find the distance between each pair of points. Round your answer to the nearest tenth if necessary.

[tex]\left( -10, -7 \right), \left( -8, 1 \right)[/tex]

[tex]
\begin{array}{l}
\sqrt{(-8 + 10)^2 + (1 + 7)^2} \\
\sqrt{2^2 + 8^2} \\
\sqrt{4 + 64} = \sqrt{68}
\end{array}
[/tex]



Answer :

Let's find the distance between the two points [tex]\((-10, -7)\)[/tex] and [tex]\((-8, 1)\)[/tex].

1. Identify the coordinates of the two points:
- First point [tex]\((x_1, y_1)\)[/tex] is [tex]\((-10, -7)\)[/tex]
- Second point [tex]\((x_2, y_2)\)[/tex] is [tex]\((-8, 1)\)[/tex]

2. Calculate the differences in the x-coordinates ([tex]\(\Delta x\)[/tex]) and y-coordinates ([tex]\(\Delta y\)[/tex]):
- [tex]\(\Delta x = x_2 - x_1 = -8 - (-10) = -8 + 10 = 2\)[/tex]
- [tex]\(\Delta y = y_2 - y_1 = 1 - (-7) = 1 + 7 = 8\)[/tex]

3. Apply the distance formula, which is:
[tex]\[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \][/tex]
Here, [tex]\(\Delta x = 2\)[/tex] and [tex]\(\Delta y = 8\)[/tex]:
[tex]\[ d = \sqrt{(2)^2 + (8)^2} = \sqrt{4 + 64} = \sqrt{68} \][/tex]

4. Evaluate the square root to find the exact distance:
[tex]\[ \sqrt{68} \approx 8.246211251235321 \][/tex]

5. Round the result to the nearest tenth:
[tex]\[ 8.2 \][/tex]

Thus, the distance between the points [tex]\((-10, -7)\)[/tex] and [tex]\((-8, 1)\)[/tex] is approximately [tex]\(8.2\)[/tex] units.