Midpoint of a segment
A fibre-optic cable between two cities needs a repeater exactly halfway along its route; the pillar of a bridge must rest exactly at the centre of the cable it supports. In both cases we are looking for the same point: the midpoint of a segment.
The midpoint formula
Calculate the coordinates of the midpoint of the segment whose endpoints are points $A(2, 4)$ and $B(6, 10)$.
Calculate the coordinates of the midpoint of the segment whose endpoints are points $A(2, 4)$ and $B(6, 10)$.
Calculate the midpoint of the segment joining points $C\left(\frac{1}{2}, 3\right)$ and $D\left(\frac{3}{2}, -1\right)$.
Calculate the midpoint of the segment joining points $C\left(\frac{1}{2}, 3\right)$ and $D\left(\frac{3}{2}, -1\right)$.
Finding the missing endpoint
If you know one endpoint $A$ and the midpoint $M$, you can solve for the other endpoint. Since $M$ is the mean of $A$ and $B$:
$$B = 2M - A$$The midpoint of segment $AB$ is $M(3, 5)$. If endpoint $A$ has coordinates $(1, 2)$, find the coordinates of endpoint $B$.
The midpoint of segment $AB$ is $M(3, 5)$. If endpoint $A$ has coordinates $(1, 2)$, find the coordinates of endpoint $B$.
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…
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 $b$.
Diagonals of a parallelogram
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…
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…
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.
Medians of a triangle
A median of a triangle is the segment that joins a vertex to the midpoint of the opposite side. By calculating that midpoint you can find the length of the median using the distance formula.
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…
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 $A$.
Dividing a segment into equal parts
By calculating successive midpoints you can split a segment into $4$, $8$... equal parts: the midpoint of $A$ and $B$ gives you the half; the midpoints of each half give you the quarters, and so on.
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…
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…
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.