To solve these problems, we will use the distance formula and the midpoint formula for the given points [tex]\((-10, 5)\)[/tex] and [tex]\((15, -11)\)[/tex].
(a) Finding the distance between the two points:
The distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a coordinate plane is given by the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting in the given coordinates:
[tex]\[
(x_1, y_1) = (-10, 5)
\][/tex]
[tex]\[
(x_2, y_2) = (15, -11)
\][/tex]
[tex]\[
d = \sqrt{(15 - (-10))^2 + (-11 - 5)^2}
\][/tex]
[tex]\[
d = \sqrt{(15 + 10)^2 + (-11 - 5)^2}
\][/tex]
[tex]\[
d = \sqrt{25^2 + (-16)^2}
\][/tex]
[tex]\[
d = \sqrt{625 + 256}
\][/tex]
[tex]\[
d = \sqrt{881}
\][/tex]
Therefore, the exact distance between the two points is [tex]\(\sqrt{881}\)[/tex].
(b) Finding the coordinates of the midpoint:
The midpoint [tex]\(M\)[/tex] of a segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the midpoint formula:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Substituting in the given coordinates:
[tex]\[
M = \left( \frac{-10 + 15}{2}, \frac{5 + (-11)}{2} \right)
\][/tex]
[tex]\[
M = \left( \frac{5}{2}, \frac{-6}{2} \right)
\][/tex]
[tex]\[
M = \left( 2.5, -3 \right)
\][/tex]
Therefore, the coordinates of the midpoint are [tex]\((2.5, -3)\)[/tex].
