To find the coordinates of vertex C in the quadrilateral ABCD, we need some additional information about the shape of ABCD. Assuming ABCD is a parallelogram (which is one of the most common assumptions), we can solve for the coordinates of C.
In a parallelogram, the diagonals bisect each other. This means that the midpoint of diagonal AC will be the same as the midpoint of diagonal BD.
Let's label the coordinates of point C as [tex]\((x, y)\)[/tex].
1. First, we find the midpoint of diagonal BD:
Vertices B and D have coordinates [tex]\((10, 7)\)[/tex] and [tex]\((2, 4)\)[/tex] respectively.
Midpoint of BD:
[tex]\[
\left( \frac{10 + 2}{2}, \frac{7 + 4}{2} \right) = \left( \frac{12}{2}, \frac{11}{2} \right) = (6, 5.5)
\][/tex]
2. Next, we find the midpoint of diagonal AC:
Vertices A and C have coordinates [tex]\((5, 7)\)[/tex] and [tex]\((x, y)\)[/tex] respectively.
Since AC shares the same midpoint as BD, we set the midpoint of AC equal to (6, 5.5):
[tex]\[
\left( \frac{5 + x}{2}, \frac{7 + y}{2} \right) = (6, 5.5)
\][/tex]
3. Now, we solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
For the x-coordinate:
[tex]\[
\frac{5 + x}{2} = 6
\][/tex]
Multiplying both sides by 2:
[tex]\[
5 + x = 12
\][/tex]
Subtracting 5 from both sides:
[tex]\[
x = 7
\][/tex]
For the y-coordinate:
[tex]\[
\frac{7 + y}{2} = 5.5
\][/tex]
Multiplying both sides by 2:
[tex]\[
7 + y = 11
\][/tex]
Subtracting 7 from both sides:
[tex]\[
y = 4
\][/tex]
Thus, the coordinates of vertex C are [tex]\((7, 4)\)[/tex].
The coordinates of vertex C are (7, 4).
