The Midpoint Formula
Marcus lives at the corner $(2, 6)$ on a city map, where each unit is one block. Aaliyah lives at $(8, 2)$. They want to meet at the spot exactly halfway between their homes. Which corner is that? A cell tower repeater placed midway between two towers, the center of a circle, and the point where a parallelogram's diagonals cross all ask for the same thing: the midpoint of a segment.
The Midpoint Formula
On a number line, the number halfway between $2$ and $8$ is $5$, the average of the two. On the coordinate plane, you do this twice: once for $x$ and once for $y$.
Why does it work? Halfway across horizontally means halfway between the two $x$-values. Halfway up or down means halfway between the two $y$-values. The midpoint is the same distance from both ends in each direction.
Calculate the coordinates of the midpoint of the line segment whose endpoints are the points $A(2, 4)$ and $B(6, 10)$.
Calculate the coordinates of the midpoint of the line segment whose endpoints are the points $A(2, 4)$ and $B(6, 10)$.
Determine the coordinates of the midpoint of the line segment whose endpoints are the points $P(-4, -6)$ and $Q(2, -8)$.
Determine the coordinates of the midpoint of the line segment whose endpoints are the points $P(-4, -6)$ and $Q(2, -8)$.
The points $A(0, 4)$ and $B(4, 0)$ form a line segment on the Cartesian plane. Calculate the coordinates of its midpoint.
The points $A(0, 4)$ and $B(4, 0)$ form a line segment on the Cartesian plane. Calculate the coordinates of its midpoint.
Fractions and Decimals
The formula doesn't care what kind of numbers you use. Just add carefully and divide by $2$.
Calculate the midpoint of the line segment joining the points $C\left(\frac{1}{2}, 3\right)$ and $D\left(\frac{3}{2}, -1\right)$.
Calculate the midpoint of the line segment joining the points $C\left(\frac{1}{2}, 3\right)$ and $D\left(\frac{3}{2}, -1\right)$.
An agricultural drone is programmed to fly in a straight line from the point $A(-15, 20)$ to the point $B(45, -10)$ (coordinates in feet). If it must drop a load of…
An agricultural drone is programmed to fly in a straight line from the point $A(-15, 20)$ to the point $B(45, -10)$ (coordinates in feet). If it must drop a load of fertilizer exactly halfway through its journey, at what coordinates will it do so?
Two telephone antennas are located at the coordinates $T_1(4.5, 8.2)$ and $T_2(-1.5, 3.4)$ on a scaled topographic plan. A repeater is to be installed exactly at the…
Two telephone antennas are located at the coordinates $T_1(4.5, 8.2)$ and $T_2(-1.5, 3.4)$ on a scaled topographic plan. A repeater is to be installed exactly at the midpoint between them. Calculate the exact coordinates where the repeater must be located.
Checking a Midpoint and Special Segments
To check whether a point $M$ is the midpoint of $AB$, compute the midpoint of $AB$ and compare. Both coordinates must match.
If even one coordinate fails, the answer is no. For example, $(-1, 5)$ is not the midpoint of $EF$ because its $y$ is wrong.
Special segments are easy. On a vertical segment, both endpoints have the same $x$, so the midpoint has that same $x$. On a horizontal segment, both endpoints have the same $y$.
Check analytically if the point $M(-1, 2)$ is the midpoint of the line segment with endpoints $E(-4, 5)$ and $F(2, -1)$.
Check analytically if the point $M(-1, 2)$ is the midpoint of the line segment with endpoints $E(-4, 5)$ and $F(2, -1)$.
Given the line segment with endpoints $G(-3, 7)$ and $H(-3, -1)$, find its midpoint and state on which axis or line parallel to the axes this segment lies.
Given the line segment with endpoints $G(-3, 7)$ and $H(-3, -1)$, find its midpoint and state on which axis or line parallel to the axes this segment lies.
Finding a Missing Endpoint
Sometimes you know one endpoint $A$ and the midpoint $M$, and you need the other endpoint $B$. Start from the formula and solve for $B$. Since $M = \dfrac{A + B}{2}$, we get $2M = A + B$, so
$$B = 2M - A \qquad\text{that is,}\qquad b_1 = 2m_1 - a_1,\quad b_2 = 2m_2 - a_2$$Here is another way to see it. To go from $A$ to $M$, you move some number of units right or left and up or down. Make the same move again and you land on $B$.
Some problems ask for the "symmetrical point of $A$ with respect to the origin." That is the reflection of $A$ through the origin (a $180^\circ$ rotation about $O$). The origin is the midpoint, so $A' = 2(0, 0) - A = (-a_1, -a_2)$: just flip both signs.
The midpoint of the line segment $AB$ is $M(3, 5)$. If the endpoint $A$ has coordinates $(1, 2)$, find the coordinates of the endpoint $B$.
The midpoint of the line segment $AB$ is $M(3, 5)$. If the endpoint $A$ has coordinates $(1, 2)$, find the coordinates of the endpoint $B$.
Find the symmetrical point of $A(5, -3)$ with respect to the origin of coordinates $O(0,0)$.
Find the symmetrical point of $A(5, -3)$ with respect to the origin of coordinates $O(0,0)$.
A laser beam strikes a lens located at the point $L(2, 5)$. If the origin point of the laser is $O(-3, 1)$ and the lens acts as a point of central symmetry, calculate…
A laser beam strikes a lens located at the point $L(2, 5)$. If the origin point of the laser is $O(-3, 1)$ and the lens acts as a point of central symmetry, calculate the coordinates of the point $P$ where the laser will impact after passing through the lens, assuming it maintains its straight trajectory.
Midpoint Problems with Unknowns
When a coordinate is a letter, set up one equation per coordinate, with the midpoint formula on one side. Then solve each equation with inverse operations.
Information like "lies on the $x$-axis" or "lies on the $y$-axis" gives you a coordinate for free. A point on the $x$-axis looks like $(x, 0)$. A point on the $y$-axis looks like $(0, y)$.
The midpoint of the line segment $AB$ is $M(a, 3)$. If we know the coordinates of the endpoints $A(-2, 5)$ and $B(6, b)$, set up the corresponding equations and…
The midpoint of the line segment $AB$ is $M(a, 3)$. If we know the coordinates of the endpoints $A(-2, 5)$ and $B(6, b)$, set up the corresponding equations and calculate the values of the parameters $a$ and $b$.
The line 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…
The line 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 $B$.
The design of a cable-stayed bridge places the anchorages of a cable at the points $A(2k, -4)$ and $B(6, k+2)$, where $k$ is an unknown parameter. The midpoint of the…
The design of a cable-stayed bridge places the anchorages of a cable at the 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 located 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.
The Midpoint of a Diameter
The center of a circle is the midpoint of every diameter. So if you know the endpoints of a diameter, the midpoint formula gives the center. The radius is the distance from the center to either endpoint.
The endpoints of a diameter of a circle are the points $P(-3, 5)$ and $Q(7, -1)$. Calculate the coordinates of the center of the circle and the measure of its radius.
The endpoints of a diameter of a circle are the points $P(-3, 5)$ and $Q(7, -1)$. Calculate the coordinates of the center of the circle and the measure of its radius.
Diagonals of a Parallelogram
This gives you two tools.
Test: compute the midpoints of the two diagonals. If they are the same point, the quadrilateral is a parallelogram. For $P(0, 0)$, $Q(5, 1)$, $R(7, 5)$, $S(2, 4)$, the midpoint of $PR$ is $(3.5, 2.5)$ and the midpoint of $QS$ is $(3.5, 2.5)$. Same point, so $PQRS$ is a parallelogram.
Find a vertex: if you know three vertices, find the midpoint of the diagonal that joins two of them, then use $2M - (\text{the other vertex})$.
Check if 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 if 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 verify if they coincide.
Three consecutive vertices of a topographic plot in the shape of a parallelogram are $A(-3, -1)$, $B(2, -2)$ and $C(5, 2)$. Using the geometric property that the…
Three consecutive vertices of a topographic plot 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 intersect at their midpoint, calculate the coordinates of the fourth vertex $D$.
Medians and Midsegments of a Triangle
A median of a triangle is a segment from a vertex to the midpoint of the opposite side. To find its length, find the midpoint first and then use the distance formula.
A segment that joins the midpoints of two sides is a midsegment.
Here the midpoints of $AC$ and $BC$ are $(0, 4)$ and $(3, 4)$. The midsegment is horizontal (parallel to the base $AB$) and has length $3$, half of $6$. Since all three midsegments are half as long as the sides they are parallel to, the triangle formed by the three midpoints has half the perimeter of the original triangle.
Given the triangle with vertices $A(-2, 1)$, $B(4, 5)$ and $C(2, -3)$, find the coordinates of the midpoint of the side $BC$ and, then, calculate the length of the…
Given the triangle with vertices $A(-2, 1)$, $B(4, 5)$ and $C(2, -3)$, find the coordinates of the midpoint of the side $BC$ and, then, calculate the length of the median starting from the vertex $A$.
Given the triangle with vertices $A(0, 0)$, $B(4, 0)$ and $C(2, 6)$, calculate the midpoints of its three sides ($M_1$, $M_2$, $M_3$). Then, show analytically by…
Given the triangle with vertices $A(0, 0)$, $B(4, 0)$ and $C(2, 6)$, calculate the midpoints of its three sides ($M_1$, $M_2$, $M_3$). Then, show analytically by calculating the distances that the perimeter of the triangle formed by the midpoints is exactly half the perimeter of the original triangle.
Dividing a Segment into Equal Parts
Take midpoints of midpoints, and you cut a segment into $4$, then $8$, then $16$ equal parts. The midpoint of $AB$ gives halves. The midpoints of each half give quarters.
If you only want the middle one of the three division points, it is just the midpoint of the whole segment. A chain of midpoints also works step by step: when a problem says "pause at the midpoint, then go from there to the next point and pause at that midpoint," compute one midpoint at a time and use the result for the next one.
In a computer-aided 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…
In a computer-aided 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 exactly equal parts. Calculate the coordinates of the central point of those three division points.
A robotic arm moves a part in a straight line from the point $P_1(-8, 4)$ to $P_2(2, -6)$, and pauses at the midpoint $M_1$. Then, it moves the part from $M_1$ to…
A robotic arm moves a part in a straight line from the point $P_1(-8, 4)$ to $P_2(2, -6)$, and pauses at the midpoint $M_1$. Then, it moves the part from $M_1$ to $P_3(10, 8)$, making a new pause at the midpoint $M_2$ of the line segment $M_1 P_3$. Calculate the exact coordinates of $M_2$.
It is desired to divide the line segment with endpoints $A(-5, 1)$ and $B(7, 9)$ into four parts of equal length using exclusively the calculation of midpoints. Detail…
It is desired to divide the line segment with endpoints $A(-5, 1)$ and $B(7, 9)$ into four parts of equal length using exclusively the calculation of midpoints. Detail the process and calculate the coordinates of the three points that divide this segment.
Midpoints with Variables and Proofs
The midpoint formula works just as well when the coordinates are expressions. The answer is then a pair of expressions. For a moving point, that means a path.
To prove that a midpoint always lies on a certain line, don't use numbers. Compute the midpoint with letters, then show its coordinates satisfy the line's equation.
Two equal point masses move in such a way that at a time $t$ (in seconds) they are at the positions $A(t, 2t-1)$ and $B(3t+2, -t+5)$. Find the expression for the…
Two equal point masses move in such a way that at a time $t$ (in seconds) they are at the positions $A(t, 2t-1)$ and $B(3t+2, -t+5)$. Find the expression for the coordinates of the center of mass (midpoint $M$) as a function of time. At what time $t$ does the center of mass cross the y-axis ($x=0$)?
A line segment on 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…
A line segment on 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 belongs to the angle bisector of the first and third quadrants (whose equation is $y = x$), regardless of the numerical values taken by $a$ and $b$.
Back to the start: Marcus at $(2, 6)$ and Aaliyah at $(8, 2)$ meet at $\left(\dfrac{2 + 8}{2}, \dfrac{6 + 2}{2}\right) = (5, 4)$.