To find the coordinates of point T, we can use the midpoint formula and knowledge of coordinate geometry. The midpoint formula states that given two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the coordinates of the midpoint [tex]\(M\)[/tex] are:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
In this problem:
- The midpoint [tex]\(M\)[/tex] is given as [tex]\((2.15, 10.7)\)[/tex].
- Point [tex]\(U\)[/tex] is given as [tex]\((13.7, 14.5)\)[/tex].
- Let the coordinates of point [tex]\(T\)[/tex] be [tex]\((x_1, y_1)\)[/tex].
We can use the midpoint formula and rearrange it to solve for [tex]\(x_1\)[/tex] and [tex]\(y_1\)[/tex].
[tex]\[
\left( \frac{x_1 + 13.7}{2}, \frac{y_1 + 14.5}{2} \right) = (2.15, 10.7)
\][/tex]
From the x-coordinates:
[tex]\[
\frac{x_1 + 13.7}{2} = 2.15
\][/tex]
Multiply both sides by 2:
[tex]\[
x_1 + 13.7 = 4.3
\][/tex]
Subtract 13.7 from both sides:
[tex]\[
x_1 = 4.3 - 13.7
\][/tex]
[tex]\[
x_1 = -9.4
\][/tex]
From the y-coordinates:
[tex]\[
\frac{y_1 + 14.5}{2} = 10.7
\][/tex]
Multiply both sides by 2:
[tex]\[
y_1 + 14.5 = 21.4
\][/tex]
Subtract 14.5 from both sides:
[tex]\[
y_1 = 21.4 - 14.5
\][/tex]
[tex]\[
y_1 = 6.9
\][/tex]
Thus, the coordinates of point [tex]\(T\)[/tex] are [tex]\((-9.4, 6.9)\)[/tex].
So,
[tex]\[ T = (-9.4, 6.9) \][/tex]
