Given Points:
The given points are:
A(5,3)
B(-2,0)
C(3,-2)

Step 1: Calculate the distances:
To determine if the given points form a right-angled triangle, we need to calculate the distances between the points.

1. Distance between A and B:
Using the distance formula, we can find the distance between points A and B:
AB = √((x2 - x1)^2 + (y2 - y1)^2)
AB = √((-2 - 5)^2 + (0 - 3)^2)
AB = √((-7)^2 + (-3)^2)
AB = √(49 + 9)
AB = √58

2. Distance between A and C:
Using the distance formula, we can find the distance between points A and C:
AC = √((x2 - x1)^2 + (y2 - y1)^2)
AC = √((3 - 5)^2 + (-2 - 3)^2)
AC = √((-2)^2 + (-5)^2)
AC = √(4 + 25)
AC = √29

3. Distance between B and C:
Using the distance formula, we can find the distance between points B and C:
BC = √((x2 - x1)^2 + (y2 - y1)^2)
BC = √((3 - (-2))^2 + (-2 - 0)^2)
BC = √((3 + 2)^2 + (-2)^2)
BC = √(25 + 4)
BC = √29

Step 2: Check for right-angled triangle:
To prove that the given points form a right-angled triangle, we need to verify if one of the distances is the hypotenuse, and the squares of the other two distances add up to the square of the hypotenuse.

1. Let's assume AB is the hypotenuse.
AB^2 = AC^2 + BC^2
(√58)^2 = (√29)^2 + (√29)^2
58 = 29 + 29
58 = 58

Since the equation holds true, we can conclude that the points A, B, and C form a right-angled triangle.

Step 3: Calculate the midpoint of the hypotenuse:
To find the midpoint of the hypotenuse, we need to calculate the average of the x-coordinates and the average of the y-coordinates.

1. Average of x-coordinates:
(x1 + x2)/2 = (5 + (-2))/2 = 3/2 = 1.5

2. Average of y-coordinates:
(y1 + y2)/2 = (3 + (-2))/2 = 1/2 = 0.5

Therefore, the midpoint of the hypotenuse is M(1.5, 0.5).

Step 4: Check the equidistance property:
To prove that the midpoint of the hypotenuse is equidistant from the vertices of the triangle, we need to calculate the distances between the midpoint and each vertex.

1. Distance between M and A:
Using the distance formula, we can find