To determine the measure of [tex]\(\overline{AG}\)[/tex], let's proceed step by step:
1. Identify given points:
[tex]\[
A = (3, 2), \quad B = (-1, 1), \quad C = (0, -3), \quad D = (4, -2)
\][/tex]
2. Find the midpoint of diagonal [tex]\(\overline{AC}\)[/tex]:
The midpoint formula for a line segment connecting points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
So, the midpoint of [tex]\(\overline{AC}\)[/tex] is:
[tex]\[
\text{Midpoint of AC} = \left( \frac{3 + 0}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{3}{2}, \frac{-1}{2} \right) = (1.5, -0.5)
\][/tex]
3. Find the midpoint of diagonal [tex]\(\overline{BD}\)[/tex]:
Similarly, the midpoint of [tex]\(\overline{BD}\)[/tex] is:
[tex]\[
\text{Midpoint of BD} = \left( \frac{-1 + 4}{2}, \frac{1 + (-2)}{2} \right) = \left( \frac{3}{2}, \frac{-1}{2} \right) = (1.5, -0.5)
\][/tex]
4. Verify point of intersection of the diagonals [tex]\(G\)[/tex]:
The midpoints of both diagonals [tex]\(\overline{AC}\)[/tex] and [tex]\(\overline{BD}\)[/tex] are the same, confirming that [tex]\(G\)[/tex] is the intersection point. Thus,
[tex]\[
G = (1.5, -0.5)
\][/tex]
5. Calculate the distance [tex]\(\overline{AG}\)[/tex]:
Using the distance formula between points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
So, the distance [tex]\(\overline{AG}\)[/tex] is:
[tex]\[
\overline{AG} = \sqrt{(1.5 - 3)^2 + (-0.5 - 2)^2} = \sqrt{(-1.5)^2 + (-2.5)^2} = \sqrt{2.25 + 6.25} = \sqrt{8.5}
\][/tex]
[tex]\[
\sqrt{8.5} \approx 2.9154759474226504
\][/tex]
6. Round to the nearest tenth:
Rounding 2.9154759474226504 to the nearest tenth gives:
[tex]\[
\overline{AG} \approx 2.9
\][/tex]
Hence, the measure of [tex]\(\overline{AG}\)[/tex], rounded to the nearest tenth, is:
[tex]\[
\boxed{2.9}
\][/tex]
