To find the midpoint of a line segment with given endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we can use the midpoint formula. The midpoint formula is given by:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Let's apply this formula to the given endpoints [tex]\(\left( -\frac{7}{3}, \frac{3}{4} \right)\)[/tex] and [tex]\(\left( \frac{5}{3}, -\frac{9}{4} \right)\)[/tex]:
1. Calculate the x-coordinate of the midpoint:
First, add the x-coordinates of the two endpoints:
[tex]\[
x_1 + x_2 = -\frac{7}{3} + \frac{5}{3}
\][/tex]
Since both terms have the same denominator, we can simply add the numerators:
[tex]\[
-\frac{7}{3} + \frac{5}{3} = \frac{-7 + 5}{3} = \frac{-2}{3}
\][/tex]
Next, divide this sum by 2 to find the x-coordinate of the midpoint:
[tex]\[
\text{Midpoint}_x = \frac{\frac{-2}{3}}{2} = \frac{-2}{3} \times \frac{1}{2} = \frac{-2}{6} = \frac{-1}{3}
\][/tex]
2. Calculate the y-coordinate of the midpoint:
Next, add the y-coordinates of the two endpoints:
[tex]\[
y_1 + y_2 = \frac{3}{4} + \left( -\frac{9}{4} \right)
\][/tex]
Since both terms have the same denominator, we can simply add the numerators:
[tex]\[
\frac{3}{4} - \frac{9}{4} = \frac{3 - 9}{4} = \frac{-6}{4} = -\frac{3}{2}
\][/tex]
Next, divide this sum by 2 to find the y-coordinate of the midpoint:
[tex]\[
\text{Midpoint}_y = \frac{-\frac{3}{2}}{2} = -\frac{3}{2} \times \frac{1}{2} = -\frac{3}{4}
\][/tex]
Therefore, the coordinates of the midpoint of the line segment with the given endpoints are:
[tex]\[
\left(-\frac{1}{3}, -\frac{3}{4}\right)
\][/tex]
In decimal form, this is:
[tex]\[
(-0.3333333333333333, -0.75)
\][/tex]
Thus, the midpoint is [tex]\((-0.33, -0.75)\)[/tex].
