Exercises: Midpoint of a line 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 line segment joining 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.

Given that the midpoint of the line segment with endpoints $A(x, 4)$ and $B(5, y)$ is $M(2, -1)$, calculate the value of the unknown coordinates $x$ and $y$.

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

On a topographic map, two telecommunications antennas are located at the coordinates $A(10, 20)$ and $B(40, -10)$. A repeater is to be installed at exactly the same distance from both, on the straight line that joins them. What will be the coordinates of the repeater?

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.

Check analytically that, if $M$ is the midpoint of the line segment with endpoints $A(0, 4)$ and $B(6, -2)$, the vector equality $\vec{AM} = \vec{MB}$ holds.

A delivery drone flies in a straight line from the warehouse $A(-50, 120)$ to the base $B(150, -40)$. Exactly at the midpoint of its journey, it drops a package with a parachute. Determine the coordinates of the drop point and the straight-line distance from the warehouse to that point.

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 $k$ so that the midpoint of the line segment with endpoints $A(k, 2)$ and $B(4, k-1)$ lies on the angle bisector of the first quadrant ($y = x$).

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

The line $r: x - 2y + 4 = 0$ intersects the coordinate axes at the points $P$ and $Q$. The line $s: 2x + y - 6 = 0$ intersects the axes at $R$ and $S$. Calculate the coordinates of the midpoint of the line segment joining the midpoints of $PQ$ and $RS$.

Find the symmetric point of $P(3, -2)$ with respect to the point $C(1, 4)$. Next, find the symmetric point of the resulting point with respect to the origin.

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

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

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

A line segment is divided into two equal parts by the point $M(2, 1)$. Given that its endpoints lie, respectively, on the lines $r: x - y - 2 = 0$ and $s: 2x + y - 5 = 0$, find the coordinates of these endpoints.

Two robotic arms move along the straight rails defined by $y = 2x + 1$ and $y = -x + 4$. Both hold a part exactly at the midpoint of the line segment joining their grippers. If the part is located at the position $M(3, 3)$, calculate the exact coordinates of each gripper.

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.

One of the diagonals of a rhombus is contained in the line $y = 2x$. The endpoints of the other diagonal are the points $A(-1, 3)$ and $C(3, 1)$. Check that the center of the rhombus lies on the given line and calculate the length of the side of the rhombus.

Given the points $A(t, 2t)$ and $B(1-t, 3)$, which depend on the parameter $t$, obtain the expression for the distance from the origin to the midpoint of $AB$. For what value of $t$ is this distance a minimum?