Find the third vertex of a triangle if its two vertices are (−1,4) and (5,2) and midpoint of one side is (0,3).

To find the third vertex of the triangle, we can use the midpoint formula. The given information is:

The two vertices of the triangle: A(−1, 4) and B(5, 2)
The midpoint of side AB: M(0, 3)
Let the third vertex of the triangle be C(x, y).

Step 1: Midpoint Formula
The midpoint M of a line segment joining two points A(x₁, y₁) and B(x₂, y₂) is given by the formula:

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Since we know the midpoint M is (0, 3), we can apply this formula to find the coordinates of the midpoint of the line segment joining A(−1, 4) and B(5, 2).

For the x-coordinate of the midpoint:

(−1 + 5) / 2 = 0

For the y-coordinate of the midpoint:

(4 + 2) / 2 = 3

These calculations confirm that the midpoint is correct, which matches the given point M(0, 3).

Step 2: Use the Midpoint Formula to Find the Third Vertex
Now, let the third vertex of the triangle be C(x, y). The midpoint of side BC is given by the midpoint formula:

M = ((x + 5) / 2, (y + 2) / 2)

We know the midpoint M is (0, 3), so we can set up the following system of equations:

(x + 5) / 2 = 0
(y + 2) / 2 = 3
Step 3: Solve the System of Equations
From equation (1):

(x + 5) / 2 = 0 Multiply both sides by 2: x + 5 = 0 x = −5

From equation (2):

(y + 2) / 2 = 3 Multiply both sides by 2: y + 2 = 6 y = 4

Step 4: Conclusion
The third vertex C of the triangle is (−5, 4).