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 $B$.

To determine the coordinates of the endpoint $B$ of a segment with known origin $A(-3, 5)$ and midpoint $M(1, -2)$, Cian wrote the following calculation on his exam:

1. a) Explain in detail which geometric point Cian actually calculated and why his algebraic approach is incorrect.
2. b) Set up the correct equations using the definition of a midpoint and find the actual coordinates of endpoint $B$.

A maritime rescue drone departs from the coastal base $A(-7, 8)$ and flies in a straight line until it locates a stranded boat at $B(3, -2)$, with coordinates given in kilometres relative to the control tower.

1. a) Without using formulas to calculate, observe the signs and numerical values of the coordinates of $A$ and $B$: give a reasoned estimate of which quadrant of the Cartesian plane the midpoint $M$ of the trajectory must lie in, and bound its abscissa and ordinate between two consecutive integers. Justify your reasoning based on the relative positions of both points.
2. b) Calculate the exact coordinates of $M$ using the arithmetic mean and check whether your previous estimate was correct.

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 ($y = x$).

The point $M(3, 2)$ is the midpoint of a segment with unknown endpoints $A$ and $B$. It is also known that endpoint $A$ lies on the line $r: y = 2x - 1$, while endpoint $B$ rests on the $x$-axis ($y = 0$).

1. a) Write the general coordinates of points $A$ and $B$ in terms of a single independent parameter for each.
2. b) Set up and solve the system of linear equations that uniquely determines the numerical coordinates of $A$ and $B$.
3. c) Check analytically whether segment $AB$ turns out to be perpendicular to line $r$ by calculating the slopes of both elements.

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

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

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

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 $D$.

On a regional map (with distances in kilometres), a courier company's regional headquarters is located at $A(1, 3)$ and a branch office at $C(9, 7)$. The company plans to build an intermediate distribution warehouse at the midpoint $M$ of segment $AC$.

1. a) Calculate the coordinates of the distribution centre $M$.
2. b) The company wants to complete a parallelogram-shaped logistics compound $ABCD$, such that $AC$ and $BD$ are its diagonals. If the second warehouse is located at $B(2, 8)$, find the coordinates of the fourth warehouse $D$ with reasoned steps, using the property shared by the diagonals of a parallelogram regarding their midpoint.
3. c) A straight motorway follows the line $r: x - 2y + 5 = 0$. Determine analytically whether the distribution centre $M$ and the warehouse $D$ have direct access to the motorway (that is, whether they lie on line $r$).

To find the coordinates of the fourth vertex $D$ of a parallelogram whose first three consecutive vertices are $A(-2, 3)$, $B(4, 1)$, and $C(6, 7)$, two students propose different approaches:

• Aoife proposes calculating the midpoint $M$ of diagonal $AC$ and using the fact that $M$ is also the midpoint of diagonal $BD$ to solve directly for the coordinates of $D$.

• Seán proposes finding the implicit equations of the line parallel to $AB$ passing through $C$ and the line parallel to $BC$ passing through $A$, and obtaining $D$ as the intersection point of both lines.

1. a) Solve the problem using the method proposed by Aoife.
2. b) Solve the problem using the method proposed by Seán.
3. c) Compare both methods: assess the number of algebraic operations and the risk of operational error in each, justifying which one is more efficient. Could Aoife's strategy be applied if the quadrilateral were a non-parallelogram trapezoid? Reason your answer based on the geometric properties of the diagonals.

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 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 $C$.

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

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.

A surveying team marking out a triangular nature reserve $ABC$ has lost the coordinates of the three original vertices, but keeps records of the coordinates of the midpoints of the three sides: $M_1(2, 1)$ (midpoint of side $AB$), $M_2(5, 4)$ (midpoint of side $BC$), and $M_3(1, 5)$ (midpoint of side $CA$).

1. a) Set up the system of equations relating the unknown coordinates of the vertices to those of the given midpoints, and find the exact coordinates of $A$, $B$, and $C$.
2. b) Determine the coordinates of the centroid $G$ of the original triangle $ABC$ and those of the centroid $G'$ of the medial triangle $M_1M_2M_3$ formed by the midpoints. What conclusion do you draw when comparing both results?
3. c) Prove algebraically that the equality $G = G'$ holds universally for any triangle in the plane, regardless of the coordinates of its vertices.

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.

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 $3x + 4y - 12 = 0$.