Exercises: Midpoint of a line segment

Calculate the coordinates of the midpoint of the line segment whose endpoints are the points $A(2, 4)$ and $B(6, 10)$.

The midpoint of the line segment $AB$ is $M(3, 5)$. If the endpoint $A$ has coordinates $(1, 2)$, find the coordinates of the endpoint $B$.

Determine the coordinates of the midpoint of the line segment whose endpoints are the points $P(-4, -6)$ and $Q(2, -8)$.

Find the symmetrical point of $A(5, -3)$ with respect to the origin of coordinates $O(0,0)$.

Calculate the midpoint of the line segment joining the points $C\left(\frac{1}{2}, 3\right)$ and $D\left(\frac{3}{2}, -1\right)$.

Check analytically if the point $M(-1, 2)$ is the midpoint of the line segment with endpoints $E(-4, 5)$ and $F(2, -1)$.

The points $A(0, 4)$ and $B(4, 0)$ form a line segment on the Cartesian plane. Calculate the coordinates of its midpoint.

Given the line segment with endpoints $G(-3, 7)$ and $H(-3, -1)$, find its midpoint and state on which axis or line parallel to the axes this segment lies.

An agricultural drone is programmed to fly in a straight line from the point $A(-15, 20)$ to the point $B(45, -10)$ (coordinates in feet). If it must drop a load of fertilizer exactly halfway through its journey, at what coordinates will it do so?

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

Two telephone antennas are located at the coordinates $T_1(4.5, 8.2)$ and $T_2(-1.5, 3.4)$ on a scaled topographic plan. A repeater is to be installed exactly at the midpoint between them. Calculate the exact coordinates where the repeater must be located.

A laser beam strikes a lens located at the point $L(2, 5)$. If the origin point of the laser is $O(-3, 1)$ and the lens acts as a point of central symmetry, calculate the coordinates of the point $P$ where the laser will impact after passing through the lens, assuming it maintains its straight trajectory.

Given the triangle with vertices $A(-2, 1)$, $B(4, 5)$ and $C(2, -3)$, find the coordinates of the midpoint of the side $BC$ and, then, calculate the length of the median starting from the vertex $A$.

The midpoint of the line segment $AB$ is $M(a, 3)$. If we know the coordinates of the endpoints $A(-2, 5)$ and $B(6, b)$, set up the corresponding equations and calculate the values of the parameters $a$ and $b$.

In a computer-aided design program, a line segment is drawn from $P_1(-4, -2)$ to $P_2(8, 6)$. The designer needs to place three points on the segment that divide it into four exactly equal parts. Calculate the coordinates of the central point of those three division points.

Check if the quadrilateral with vertices $A(-1, 2)$, $B(3, 4)$, $C(4, 0)$ and $D(0, -2)$ is a parallelogram. To do this, calculate the midpoints of its two diagonals and verify if they coincide.

The design of a cable-stayed bridge places the anchorages of a cable at the points $A(2k, -4)$ and $B(6, k+2)$, where $k$ is an unknown parameter. The midpoint of the cable must rest on a vertical pillar located exactly on the x-axis (that is, its y-coordinate is $y=0$). Calculate the value of $k$ and the exact coordinates of the support point.

Three consecutive vertices of a topographic plot in the shape of a parallelogram are $A(-3, -1)$, $B(2, -2)$ and $C(5, 2)$. Using the geometric property that the diagonals of a parallelogram intersect at their midpoint, calculate the coordinates of the fourth vertex $D$.

The line segment $AB$ has its midpoint at $M(4, 3)$. Knowing that the endpoint $A$ lies on the x-axis and the endpoint $B$ lies on the y-axis, set up the corresponding equations and determine the exact coordinates of $A$ and $B$.

A robotic arm moves a part in a straight line from the point $P_1(-8, 4)$ to $P_2(2, -6)$, and pauses at the midpoint $M_1$. Then, it moves the part from $M_1$ to $P_3(10, 8)$, making a new pause at the midpoint $M_2$ of the line segment $M_1 P_3$. Calculate the exact coordinates of $M_2$.

Two equal point masses move in such a way that at a time $t$ (in seconds) they are at the positions $A(t, 2t-1)$ and $B(3t+2, -t+5)$. Find the expression for the coordinates of the center of mass (midpoint $M$) as a function of time. At what time $t$ does the center of mass cross the y-axis ($x=0$)?

It is desired to divide the line segment with endpoints $A(-5, 1)$ and $B(7, 9)$ into four parts of equal length using exclusively the calculation of midpoints. Detail the process and calculate the coordinates of the three points that divide this segment.

Given the triangle with vertices $A(0, 0)$, $B(4, 0)$ and $C(2, 6)$, calculate the midpoints of its three sides ($M_1$, $M_2$, $M_3$). Then, show analytically by calculating the distances that the perimeter of the triangle formed by the midpoints is exactly half the perimeter of the original triangle.

A line segment on the Cartesian plane has generic endpoints $A(a, b)$ and $B(b, a)$, with $a \neq b$. Prove algebraically that the midpoint of this segment always belongs to the angle bisector of the first and third quadrants (whose equation is $y = x$), regardless of the numerical values taken by $a$ and $b$.