Exercises: Midpoint of a segment

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

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

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

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

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

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

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

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

Cian has to calculate the midpoint of the segment joining points $A(-4, 7)$ and $B(2, -1)$. In his notebook, he writes the following:

Identify the conceptual error Cian made, explaining which other analytic geometry formula or procedure he confused it with. Then, explain why his result is visually impossible considering the position of the points, and calculate the correct solution.

Consider the line segment with endpoints $A(-9, 2)$ and $B(3, 10)$.

1. a) Before performing any algebraic calculation, reasonably estimate in which quadrant of the Cartesian plane the midpoint $M$ must lie and predict the sign of its abscissa and its ordinate, comparing the absolute values of the endpoints.
2. b) Apply the midpoint formula to calculate the exact coordinates of $M$.
3. c) Check whether the obtained result confirms your estimate and justify why the abscissa of $M$ is at exactly the same distance from $-9$ as from $3$.

A rescue boat sails in a straight line from point $A(-15, 20)$ to point $B(45, -10)$ (coordinates in metres). If it needs to drop a marker buoy exactly halfway along its route, 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 centre of the circle and the measure of its radius.

Two mobile phone masts are located at coordinates $T_1(4.5; 8.2)$ and $T_2(-1.5; 3.4)$ on a scaled topographical map. A repeater is to be installed exactly at the midpoint between them. Calculate the exact coordinates where the repeater should be located.

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

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

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

In a computer 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 equal parts. Calculate the coordinates of the central point of these three division points.

You can do it on paper and send a photo with your answer.

Check whether 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 check whether they coincide.

You want to construct a straight line segment in the Cartesian plane such that its exact midpoint is $M(4, -3)$.

1. a) Determine the coordinates of endpoints $A$ and $B$ knowing that point $A$ is obligatorily located on the x-axis ($y = 0$) and point $B$ on the y-axis ($x = 0$).
2. b) Set up the corresponding system of equations and justify with reasoning whether this pair of points $(A, B)$ is unique or if there can be more solutions satisfying both conditions.
3. c) Write the coordinates of another, different pair of endpoints, $C$ and $D$, whose midpoint is still $M$, but with the condition that neither of them belongs to the coordinate axes.

In the Sierra Nevada National Park, the mountain rescue service needs to install a telecommunications repeater at the exact midpoint of the line connecting Refuge A, located at coordinates $(12, 18)$, and Refuge B, located at $(36, 6)$ (coordinates expressed in kilometres).

1. a) Model the situation in the plane and calculate the location coordinates of the main repeater.
2. b) For security reasons in the link protocol, two secondary beacons must also be placed: beacon $S_1$ at the midpoint between Refuge A and the main repeater, and beacon $S_2$ at the midpoint between the main repeater and Refuge B. Obtain the coordinates of both beacons using the method of successive midpoints.
3. c) Interpret the division of the path: into how many parts of equal length has the original section between both refuges been divided?

The design of a cable-stayed bridge places the anchors of a cable at 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 situated 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 topographical plot of land 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 bisect each other, calculate the coordinates of the fourth vertex $D$.

The 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 component in a straight line from point $P_1(-8, 4)$ to $P_2(2, -6)$, pausing at the midpoint $M_1$. Then, it moves the component from $M_1$ to $P_3(10, 8)$, making another pause at the midpoint $M_2$ of line segment $M_1 P_3$. Calculate the exact coordinates of $M_2$.

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

We want to divide the segment with endpoints $A(-5, 1)$ and $B(7, 9)$ into four equal parts using only midpoint calculations. Detail the process and calculate the coordinates of the three points that divide the segment.

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

A segment in 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 lies on the bisector of the first and third quadrants (whose equation is $y = x$), regardless of the numerical values taken by $a$ and $b$.

Three consecutive vertices of a plot in the shape of a parallelogram $ABCD$ are known: $A(-2, 1)$, $B(4, 3)$, and $C(6, 9)$. To find the coordinates of the fourth vertex $D$, two draftspersons propose different approaches:

• Method 1: Use the equipollence of vectors by equating the components of $\vec{AB}$ and $\vec{DC}$.

• Method 2: Calculate the midpoint of the known diagonal $AC$ and use the fact that it must coincide with the midpoint of the other diagonal, $BD$.

1. a) Fully develop both methods and verify that they lead to the same numerical result for $D$.
2. b) If the topographical map also required finding the centre of symmetry of the plot (the central point where the diagonal fences intersect), compare the efficiency of both methods and justify which of them saves steps.
3. c) If it were not specified that vertices $A$, $B$, and $C$ are in consecutive alphabetical order, reason how many different parallelograms could be constructed with those same three points as vertices.

In a triangle with vertices $A(-1, 3)$, $B(5, 7)$, and $C(3, -1)$:

1. a) Calculate the coordinates of the midpoint $M$ of side $BC$ and obtain the components of the vector going from vertex $A$ to that midpoint (the segment defining the median of the triangle).
2. b) The centroid $G$ of a triangle is the point located on each median at two thirds of the distance from the corresponding vertex to the opposite side, satisfying the vector relation $\vec{AG} = 2\cdot \vec{GM}$. Calculate the exact coordinates of $G$ for this triangle and algebraically prove that these coordinates coincide with the arithmetic mean of the coordinates of its three vertices: $\left(\frac{x_A+x_B+x_C}{3},\ \frac{y_A+y_B+y_C}{3}\right)$.
3. c) If we complete a parallelogram having segments $AB$ and $AC$ as adjacent sides, reason without needing to calculate distances whether the centroid $G$ can coincide with the intersection point of the diagonals of that parallelogram.