Points (5; 2), (2; -3), (2,1) are the midpoints of the sides of the triangle. Find the coordinates of its vertices.

The line connecting the midpoints of the two sides is parallel to the third side of the triangle.

A1 (5; 2), B1 (2; -3), C1 (2,1)

In order to find the coordinates of the midpoint of the segment, we need the formulas:

1.x = (x1 + x2) / 2

2.y = (y1 + y2) / 2

Therefore, we get a system of equations:

1. (x1 + x2) / 2 = 5 | · 2

x1 + x2 = 10

2. (x2 + x3) / 2 = 2 | · 2

x2 + x3 = 4

3. (x1 + x3) / 2 = 2 | · 2

x1 + x3 = 4

Next, we need to add all three equations:

2×1 + 2×2 + 2×3 = 18 | : 2

x1 + x2 + x3 = 9

Now from the resulting equation, subtract each equation of the system:

1.×3 = -1

2.×1 = 5

3.×2 = 5

Similarly, we compose and solve the second system of equations using the formula:

1. (y1 + y2) / 2 = 2 | · 2

y1 + y2 = 4

2. (y2 + y₃) / 2 = – 3 | · 2

y2 + y3 = – 6

3. (y1 + y3) / 2 = 1 | · 2

y1 + y3 = 2

We add all three equations, we get:

2y1 + 2y2 + 2y3 = 0 | : 2

y1 + y2 + y3 = 0

Now we subtract each equation of the system from the obtained equation:

1.y3 = – 4

2.y1 = 6

3.y2 = – 2

The obtained coordinates of points A, B, C and will be the coordinates of the vertices of the triangle ABC. (5; 6), (5; – 2), (- 1; – 4).

Answer: (5; 6), (5; – 2), (- 1; – 4).