Midpoint of a segment

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

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

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

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

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

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 $b$.

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 $B$.

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 $D$.

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.

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.

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 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.

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.

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.

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.