Exercises: Midpoint of a segment

Calculate the coordinates of the midpoint of the line segment with endpoints $A(2, -4)$ and $B(-6, 8)$.

The point $M(1, 3)$ is the midpoint of the segment connecting the points $A(-2, 5)$ and $B$. Determine the coordinates of the endpoint $B$.

The points $P(-3, -1)$ and $Q(5, 7)$ are the endpoints of the diameter of a circle. Find the coordinates of its center.

Find the midpoint of the segment determined by the points of intersection of the line $2x + 3y - 12 = 0$ with the coordinate axes.

The opposite vertices of a square are located at the points $A(1, 1)$ and $C(7, 5)$. Determine the coordinates of the center of the square.

The midpoints of the sides of a triangle are $M_1(1, 2)$, $M_2(3, 4)$ and $M_3(-1, 5)$. Set up a system of equations and calculate the coordinates of the three vertices of the triangle $A, B$ and $C$.

Three consecutive vertices of a parallelogram are $A(-2, 1)$, $B(4, 3)$ and $C(6, -1)$. Using the property that the diagonals of a parallelogram bisect each other, find the coordinates of the fourth vertex $D$.

Determine the value of the parameter $k$ such that the midpoint of the segment with endpoints $A(-k, 2)$ and $B(4, k-1)$ belongs to the angle bisector of the first quadrant ($y = x$).

A laser beam leaves the emitter $S(2, 8)$ and must hit the receiver $T(10, 8)$ after bouncing off a flat mirror located on the line $y = 5$. Check whether the midpoint of segment $ST$ coincides with the point of impact on the mirror.

Find the equation of the median corresponding to vertex $A$ in the triangle with vertices $A(1, -3)$, $B(5, 1)$ and $C(-3, 5)$.

Calculate the coordinates of the point symmetric to $A(1, 5)$ with respect to the line $r: 2x - y - 3 = 0$.

Determine the equation of the locus of the midpoints of all segments connecting the origin of coordinates to the points on the line $3x + 4y - 12 = 0$.

Prove analytically that the midpoints of the sides of the quadrilateral with vertices $A(0,0)$, $B(6,2)$, $C(4,8)$ and $D(-2.4)$ form a parallelogram (Varignon's parallelogram) and calculate its area.

The centroid of a triangle is the point $G(2, 1)$. If two of its vertices are $A(-1, 3)$ and $B(4, -2)$, find the coordinates of the third vertex $C$ based on the property of the centroid with respect to the midpoints of the sides.