Answer:
To find the coordinates of point C, we can use the fact that triangle ABC is an isosceles triangle with base AB. This means that the lengths of AC and BC are equal.
Let's start by finding the distance between points A and B. We can use the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For points A(-2, 3) and B(4, -5), the distance AB is:
AB = sqrt((4 - (-2))^2 + (-5 - 3)^2)
= sqrt(6^2 + (-8)^2)
= sqrt(36 + 64)
= sqrt(100)
= 10
Since triangle ABC is isosceles, the distance AC is also 10 units.
Now, let's consider the sum of the coordinates of point C. We know that the sum of the x-coordinates and the sum of the y-coordinates of point C is equal to 7.
So, we have:
x-coordinate of C + y-coordinate of C = 7
Let's assume the coordinates of point C are (a, b). Using the above equation, we can write:
a + b = 7
Since triangle ABC is isosceles, the distance AC is equal to 10. We can use the distance formula again to find the distance AC:
AC = sqrt((a - (-2))^2 + (b - 3)^2)
= sqrt((a + 2)^2 + (b - 3)^2)
Since AC = 10, we can write:
sqrt((a + 2)^2 + (b - 3)^2) = 10
Now, we have two equations:
a + b = 7
sqrt((a + 2)^2 + (b - 3)^2) = 10
Solving these equations simultaneously will give us the coordinates of point C. Let's solve them together.