The midpoint of a segment
A cable car needs a support tower exactly halfway between the base station and the summit. That "halfway point" has a name: the midpoint of the segment. And since it is just another point, it can also be calculated using coordinates.
The midpoint formula
When determining the midpoint $M$ of the segment with endpoints $C(-4, 2)$ and $D(6, 8)$, a classmate subtracted the coordinates of the endpoints and divided by two:…
When determining the midpoint $M$ of the segment with endpoints $C(-4, 2)$ and $D(6, 8)$, a classmate subtracted the coordinates of the endpoints and divided by two: $M\left(\frac{6 - (-4)}{2}, \frac{8 - 2}{2}\right) = (5, 3)$. Correct this common error and find the correct coordinates of the midpoint.
Calculate the coordinates of the midpoint of the line segment with endpoints $A(4, -2)$ and $B(-6, 8)$.
Calculate the coordinates of the midpoint of the line segment with endpoints $A(4, -2)$ and $B(-6, 8)$.
Finding the missing endpoint
If you know one endpoint $A$ and the midpoint $M$, you can work out the other endpoint $B$: since $M$ is the mean of $A$ and $B$, then $B = 2M - A$.
Point $M(1, 1)$ is the midpoint of the segment with endpoints $A(-2, 3)$ and $B$. Calculate the coordinates of endpoint $B$.
Point $M(1, 1)$ is the midpoint of the segment with endpoints $A(-2, 3)$ and $B$. Calculate the coordinates of endpoint $B$.
A bridge structure features a diagonal beam running from anchor $A(2, 1)$ to node $B(14, 9)$, expressed in metres. A strain sensor needs to be installed exactly at the…
A bridge structure features a diagonal beam running from anchor $A(2, 1)$ to node $B(14, 9)$, expressed in metres. A strain sensor needs to be installed exactly at the centre of the beam. Calculate the coordinates of the sensor.
Symmetry about a point
Find the coordinates of the reflection of $P(5, -2)$ in point $Q(0, 4)$.
Find the coordinates of the reflection of $P(5, -2)$ in point $Q(0, 4)$.
Applications: diagonals and centres
The midpoint lies behind many geometric properties: the centre of a circle is the midpoint of any of its diameters; in a parallelogram, the two diagonals intersect at their common midpoint.
The line segment joining points $P(-4, 7)$ and $Q(8, 3)$ is the diameter of a circle. Calculate the coordinates of the centre of the circle and the exact length of its…
The line segment joining points $P(-4, 7)$ and $Q(8, 3)$ is the diameter of a circle. Calculate the coordinates of the centre of the circle and the exact length of its radius.
Three of the vertices of a parallelogram $ABCD$ are $A(-1, -2)$, $B(4, -1)$ and $C(5, 3)$. Knowing that the diagonals of a parallelogram bisect each other, calculate the…
Three of the vertices of a parallelogram $ABCD$ are $A(-1, -2)$, $B(4, -1)$ and $C(5, 3)$. Knowing that the diagonals of a parallelogram bisect each other, calculate the coordinates of the fourth vertex $D$.
Given any quadrilateral with vertices at $A(-4, 2)$, $B(2, 6)$, $C(6, 0)$ and $D(0, -6)$, calculate the coordinates of the midpoints of its four sides. Then, prove…
Given any quadrilateral with vertices at $A(-4, 2)$, $B(2, 6)$, $C(6, 0)$ and $D(0, -6)$, calculate the coordinates of the midpoints of its four sides. Then, prove mathematically that the new quadrilateral formed by joining these midpoints is always a parallelogram.