Midpoint of a Segment
Wilkin and Yaritza live at opposite ends of the Autopista Duarte, he at a point $A$ and she at a point $B$ on a grid map. They want to meet exactly halfway, at a corner store that is the same distance from both of them. How do you calculate the coordinates of that spot? That point is the midpoint, the simplest tool in analytic geometry and one of the most often used hidden inside other problems: diagonals that intersect, centers of figures, symmetries. Almost every time a problem mentions a "center" or a "symmetric" point, a midpoint is behind it.
The midpoint formula
It makes sense: to be halfway horizontally, the $x$-coordinate of $M$ lies in the middle of those of $A$ and $B$; the same goes for $y$.
Calculate the coordinates of the midpoint of the line segment with endpoints $A(2, -4)$ and $B(-6, 8)$.
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 $B$.
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 $B$.
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…
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 midpoint in geometric figures
The points $P(-3, -1)$ and $Q(5, 7)$ are the endpoints of the diameter of a circle. Find the coordinates of its center.
The points $P(-3, -1)$ and $Q(5, 7)$ are the endpoints of the diameter of a circle. Find the coordinates of its center.
The opposite vertices of a square are located at the points $A(1, 1)$ and $C(7, 5)$. Determine the coordinates of the center of the square.
The opposite vertices of a square are located at the points $A(1, 1)$ and $C(7, 5)$. Determine the coordinates of the center 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.
Find the midpoint of the segment determined by the points of intersection of the line $2x + 3y - 12 = 0$ with the coordinate axes.
Diagonals of a parallelogram
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…
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 $D$.
The midpoints of a quadrilateral: Varignon's parallelogram
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…
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 triangle, the median, and the centroid
In the figure, $M$ is the midpoint of side $AB$ and $CM$ is the median from $C$.
To locate the centroid you use its formula: it is the average of the coordinates of the three vertices.
Find the equation of the median corresponding to vertex $A$ in the triangle with vertices $A(1, -3)$, $B(5, 1)$ and $C(-3, 5)$.
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…
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.
Extension: reconstructing a triangle from midpoints
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…
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 $C$.
The point symmetric about a line
It also works for loci: if a point moves along a line, its midpoint with the origin moves along another line (you find it by expressing $x$ and $y$ of the midpoint and substituting into the original equation).
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…
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 $r: 2x - y - 3 = 0$.
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 $3x + 4y - 12 = 0$.
Determine the equation of the locus of the midpoints of all segments connecting the origin of coordinates to the points on the line $3x + 4y - 12 = 0$.
Back to the beginning: the corner store of Wilkin and Yaritza is at the midpoint $M$ of their two homes, which is obtained by averaging the coordinates of $A$ and $B$.