Exercises: Midpoint of a line segment

When determining the midpoint $M$ of the segment with endpoints $C(-4, 2)$ and $D(6, 8)$, a classmate subtracted the coordinates of the endpoints and divided by two: $M\left(\frac{6 - (-4)}{2}, \frac{8 - 2}{2}\right) = (5, 3)$. Correct this common error and find the correct coordinates of the midpoint.

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

Find the midpoint of the line segment whose endpoints are points $C(-3, -5)$ and $D(-1, -9)$.

Determine the coordinates of the midpoint of the line segment formed by points $E\left(\frac{1}{2}, \frac{3}{4}\right)$ and $F\left(\frac{5}{2}, -\frac{1}{4}\right)$.

Point $M(1, 1)$ is the midpoint of the segment with endpoints $A(-2, 3)$ and $B$. Calculate the coordinates of endpoint $B$.

Find the coordinates of the reflection of $P(5, -2)$ in point $Q(0, 4)$.

Check mathematically whether point $M(2, -1)$ is truly the midpoint of the line segment joining points $R(7, -5)$ and $S(-3, 3)$.

Calculate the coordinates of the point that bisects the line segment joining the origin to point $P(-14, 22)$.

A cable car connects the base station located at point $B(100, 200)$ to the mountain peak at point $P(500, 800)$ on a coordinate plane. Find the coordinates of the intermediate support tower that must be built exactly halfway.

Marcos has to calculate the coordinates of the midpoint of the segment with endpoints $A(-6, 8)$ and $B(2, -4)$. In his notebook, he writes the following solution:

Identify what conceptual mistake Marcos has made (what other analytic geometry formula or concept did he confuse it with?), explain with reasons why his result does not make sense visually when looking at the position of both points in the Cartesian plane, and calculate the correct solution.

The State Meteorological Agency (AEMET) plans to locate an intermediate rain gauge station exactly at the midpoint between two automatic weather stations situated in the municipalities of Almazán, $A(14, 32)$, and Sigüenza, $B(26, -4)$, where the coordinates are given in kilometres with respect to the regional coordination centre.

1. a) Without doing the exact numerical calculation of the formula, make a reasoned estimate: indicate between which integer values the $x$-coordinate and the $y$-coordinate of the new station must lie, and deduce in which quadrant of the map it will be located.
2. b) Calculate the exact coordinates of the midpoint $M$.
3. c) Check whether your initial estimate was correct and justify whether the result obtained is consistent with the positions of both municipalities.

On a town plan, two underground stations are located at coordinates $A(-2,5; 4,8)$ and $B(7,3; -1,2)$. Engineers need to drill a ventilation shaft at the exact midpoint between both stations. Calculate the exact coordinates of this shaft.

The line segment joining points $P(-4, 7)$ and $Q(8, 3)$ is the diameter of a circle. Calculate the coordinates of the centre of the circle and the exact length of its radius.

Three of the vertices of a parallelogram $ABCD$ are $A(-1, -2)$, $B(4, -1)$ and $C(5, 3)$. Knowing that the diagonals of a parallelogram bisect each other, calculate the coordinates of the fourth vertex $D$.

The midpoint of the segment whose endpoints are $A(3m, n-2)$ and $B(m+4, 3n)$ is point $M(7, 5)$. Set up a system of equations and calculate the values of parameters $m$ and $n$.

In a video game, a character goes through a teleportation portal that functions as a point of central symmetry located at $M(15, 20)$. If the character enters the portal from position $C(5, 8)$, at what coordinates will they reappear? If after reappearing they walk in a straight line to position $D(30, 40)$, what is the midpoint of this final path?

A bridge structure features a diagonal beam running from anchor $A(2, 1)$ to node $B(14, 9)$, expressed in metres. A strain sensor needs to be installed exactly at the centre of the beam. Calculate the coordinates of the sensor.

A triangle has vertices at points $A(0, 0)$, $B(6, 0)$ and $C(2, 8)$. Calculate the coordinates of the midpoints of its three sides. Then, find the perimetre of the new triangle formed by these midpoints and check that it is exactly half the perimetre of the original triangle $ABC$.

In a two-dimensional astronomical model, a comet is detected at position $P_1(-300, 450)$ and several days later at position $P_2(150, -150)$ (coordinates in millions of kilometres). Find the coordinates of the point that lies exactly halfway between these two observations. Then, calculate the distance from the origin (where the Sun is located) to this midpoint.

A freight transport operator has its central hub located at coordinate $M(3, -2)$ on a regional grid.

1. a) A broken-down delivery van is at point $A(-5, 4)$. Determine algebraically the exact coordinates of point $B$ where an assistance vehicle must be located so that the hub $M$ is exactly at the midpoint between both.
2. b) Now approach the problem in a reverse, open-ended way: find the coordinates of two other distinct points, $C$ and $D$ (with integer coordinates and different from the previous ones), such that $M(3, -2)$ is still their midpoint. Explain what general mathematical relationship the sums of their abscissas and their ordinates must satisfy for any pair of points to be valid.

The regional emergency medical service must set up a new Basic Life Support ambulance station to serve two isolated health centres, located at points $A(-14, 26)$ and $B(6, -8)$, with coordinates in kilometres.

1. a) For territorial equity, station $M$ must be placed at the midpoint of the segment joining both centres. Calculate the coordinates of station $M$.
2. b) The regional protocol establishes that an ambulance station can only provide additional emergency coverage to a nursing home located at $R(-10, 15)$ if the straight-line distance from the station to the nursing home is less than $10\text{ km}$. Determine the components of the vector $\vec{MR}$, calculate its magnitude, and justify with reasons whether the nursing home will be within the regulatory coverage radius.

Given any quadrilateral with vertices at $A(-4, 2)$, $B(2, 6)$, $C(6, 0)$ and $D(0, -6)$, calculate the coordinates of the midpoints of its four sides. Then, prove mathematically that the new quadrilateral formed by joining these midpoints is always a parallelogram.

The endpoints of a segment are the points $A(k^2, 3k)$ and $B(-4, k-8)$. Determine the value or values of the parameter $k$ for which the midpoint of this segment coincides exactly with the origin of coordinates.

A laser cutting machine makes a straight cut on a metal plate from point $P(-8, 10)$ to point $Q(12, -6)$. The system needs to place exactly three equally spaced rivets along this cut (that is, dividing the segment into 4 equal parts). Calculate the coordinates of the three rivet points by repeatedly using the concept of a midpoint.

A point $P(-2, 5)$ is reflected in point $M_1(1, 3)$ to obtain a point $P'$. Subsequently, point $P'$ is reflected in point $M_2(4, -1)$ to obtain a final point $P''$. Calculate the coordinates of $P''$ and prove that the translation vector $\vec{PP''}$ is exactly twice vector $\vec{M_1M_2}$.

In a physics experiment, four identical point masses are placed at coordinates $A(-4, 2)$, $B(2, 6)$, $C(4, -2)$ and $D(-2, -4)$. To find the centre of mass of the entire system, first calculate the midpoint of line segment $AB$ and the midpoint of line segment $CD$. Then, calculate the midpoint of the line segment joining the previous two midpoints. What are the coordinates of the centre of mass of the system?

The centre of a rectangle is located at the point $M(3, 2)$. The coordinates of two of its adjacent vertices are known: $A(-1, -1)$ and $B(7, -1)$. Calculate the coordinates of the other two vertices, $C$ and $D$. Finally, check analytically that the two diagonals of the rectangle have exactly the same length.

The reflection of $A(x, y)$ in point $M(2, -3)$ is $A'$. In turn, the reflection of $A'$ in point $N(-1, 4)$ is $A''$. If we know that the final coordinates are $A'' = (5, 7)$, set up the corresponding equations and determine the coordinates of the original point $A$.

Let $A(-3, 4)$ be a fixed point in the plane. A second point $B(x, y)$ moves freely, remaining on the line with equation $y = 2x$. Express the coordinates of the midpoint $M$ of line segment $AB$ in terms of $x$. From there, deduce the algebraic relationship (equation) satisfied by the coordinates of point $M$.

On an irrigated farm in Castilla-La Mancha, a plot shaped like a parallelogram $ABCD$ is marked out. Three of its vertices are recorded with GPS: $A(-3, 2)$, $B(5, 4)$, and $C(7, -2)$. To find the position of the fourth vertex $D$, two technicians propose different strategies:

• Technician 1: Find the midpoint $M$ of the known diagonal $AC$ and, relying on the property that the diagonals bisect each other at their midpoint, solve for $D$ knowing that $M$ is also the midpoint of the diagonal $BD$.

• Technician 2: Equate the free vectors of opposite parallel sides (for example, $\vec{AB} = \vec{DC}$, satisfying $B - A = C - D$) and directly solve for the coordinates of $D$.

1. a) Apply both methods step by step to find the coordinates of $D$ and verify that both paths lead to the same result.
2. b) Compare the two procedures: which one requires fewer calculation steps? Furthermore, if the irrigation community needs to know the exact location of the geometric centre of the plot to drill a supply well, which of the two methods provides that information directly? Justify your answer.

A Salvamento Marítimo helicopter takes off from coastal base $A(-12, 18)$ to assist a fishing boat sending a distress signal from point $B(8, -6)$ (coordinates given in nautical miles). Coordination instructions order an intermediate life raft $Q$ to be anchored exactly at the midpoint of the route between the base and the fishing boat.

1. a) Model the situation by calculating the exact coordinates of the intermediate raft $Q$.
2. b) Suppose that, due to strong winds and an orientation error, the helicopter overshoots raft $Q$ continuing along the same straight line and travels beyond $Q$ exactly the same distance that separates base $A$ from raft $Q$, reaching point $A'$ (that is, the reflection of $A$ across $Q$). Calculate the coordinates of $A'$ using the expression $A' = 2Q - A$.
3. c) Observing the numerical result obtained, a duty officer claims: "Any point reflected across the midpoint of a segment necessarily coincides with the other endpoint of the segment." Prove algebraically (using generic letters $A(a_1, a_2)$ and $B(b_1, b_2)$) whether the officer's statement is a universal mathematical property or if it was just a coincidence of the coordinates in this problem.