Midpoint of a segment
The midpoint is the simplest tool in coordinate geometry and, at the same time, one of those most often hidden inside other problems: diagonals that intersect, centres of figures, symmetries... Almost whenever a question talks about a "centre" or a "symmetric point", there is a midpoint behind it.
The midpoint formula
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 centre.
The points $P(-3, -1)$ and $Q(5, 7)$ are the endpoints of the diameter of a circle. Find the coordinates of its centre.
The opposite vertices of a square are located at the points $A(1, 1)$ and $C(7, 5)$. Determine the coordinates of the centre 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 centre 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 $D$.
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
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.
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 $3x + 4y - 12 = 0$.