Midpoint of a Segment
Ximena and Diego agree to meet on the Metro: she boards at station $(2, 4)$ and he boards at station $(10, 8)$ on a grid map. They want to meet exactly halfway. At what point on the map do they meet? That point is the midpoint, and it shows up hidden in many problems: diagonals that intersect, centers of figures, symmetries. At the end of the page we come back to this question.
The Midpoint Formula
It makes sense: halfway along $x$ is the average of the two $x$ positions, and 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 $D$.
Varignon Parallelogram: Parallelism and Area
If you join the midpoints of the sides of any quadrilateral, you always get a parallelogram (the Varignon parallelogram). To prove this with coordinates you need two tools: checking that two sides are parallel and calculating an area.
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
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 $C$.
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.
The Symmetric Point About a Line
Locus of midpoints
If a point $P(x, y)$ moves along a line and $M$ is the midpoint between the origin and $P$, then $M = (x/2, y/2)$. You solve $x = 2m_x$, $y = 2m_y$, substitute into the equation of the line, and you get the equation that $M$ satisfies.
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: Ximena at $(2, 4)$ and Diego at $(10, 8)$ meet at $\left(\dfrac{2+10}{2}, \dfrac{4+8}{2}\right) = (6, 6)$.