Midpoint Formula

One of the most important theorems concerning coordinate geometry is the midpoint theorem formula. From problem-solving in schools to chart plotting and architecture, it is used everywhere. It can be said that the midpoint theorem formula is one of the solid bases on which entire geometry rests, and like the rest of mathematics, it too is devoid of any exceptions, and hence grants a sense of certainty in all the aforementioned fields. Much of our modern-day applied geometry is used in cartography, architecture, and structural problem-solving. Aircraft get better and better as we adjust the geometry and hence, the aerodynamics, buildings get stronger and more resistant to natural disasters as we learn to master the shapes used in construction, maps are plotted and distances measured without the use of any instruments, simply because we know the rules of math, and the fact that it has no exceptions. Whether we are dealing with intricate maps or simply trying to find what length a particular side of a given triangle is, it is certain that coordinate geometry is the way forward. 

What is Midpoint Theorem?

The midpoint theorem formula is extremely useful for circles, triangles, ellipses, Hyperbole, and all the General polygons. But our first concern should be to talk about the midpoint theorem formula as it applies to a straight-line segment on a cartesian plane. 

For a cartesian plane, a line segment is defined by two endpoints on the coordinate plane whose coordinates are known. The midpoint of this line is just the halfway between these endpoints and we can find its location using the Midpoint Theorem, which states:

“The x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints. In the same way, the y-coordinate is nothing more than just an average of the y-coordinates of the endpoints.”

Midpoint Formula

If we want to find the midpoint of the straight line in a graph, we must use this formula to find the midpoint that will help us to find the coordinates of the endpoint of the given line. Suppose the endpoints of the line is (x1, y1) and (x2, y2) then the formula to find the midpoint is given as 

(x,y) = \[\frac{(x1 + x2)}{2}\] , \[\frac{(y1 + y2)}{2}\]

Hence we can predict accurately where the midpoint of any line segment is located once we know the coordinates of its endpoints. It is to be noted that the reverse is not possible, however. We cannot predict the location of the endpoints, as there are a large number of possible number combinations that can give the exact same ratio. Visually, the orientation of the line segment can be changed while keeping the midpoint fixed, hence coordinates of at least one endpoint, as well as the midpoint, should be known if we are to accurately predict the other endpoint. 

 Apart from cartesian planes, The theorem is widely used in triangles. The Midpoint theorem for triangles explains that if we draw a line from the center of one side of a triangle, parallelly to another side, it will bisect the third side of the triangle. Its inverse is also true. A line segment joining the midpoints of two sides of a triangle will be parallel to the third side and will measure half of that third side. 

Solved Examples

Question 1: Prove that the midpoint formula of XY =  (1/2 *  QR) if a triangle PQR, with X and Y as the midpoints of the sides PQ and QR is taken. 

Note: The line XY is to be parallel to the line QR, and exactly half of its length.

Solution:

Extend the line segment XY and produce it to Z such that, YZ=XY

In the triangle, PXY, and also  the triangle RZY

YR= PY —–   (given)

∠RYZ = ∠PYX {vertically opposite angles}

YZ = XY { by construction}

hence,

△ RZY ≅  △ PXY {by SAS}

Therefore,

∠RZY = ∠PXY {by c.p.c.t.}

∠ZRY= ∠XPY    {by c.p.c.t.}

and RZ = PX {by c.p.c.t.}

The angles, ∠RZY, and ∠PXY are the alternate interior angles. Assume  RZ and PQ as two lines which are intersected by the transversal XZ.

In a similar way, ∠ZRY and ∠XPY are alternate interior angles.  Assume RY and PQ are the two lines which are intersected by the transversal PR.

Therefore, RZ ∥ PQ

So, RZ ∥ QX

and RZ = QX  {since QX = PX, it is proved that RZ = PX}

Thus, QXZR forms a parallelogram.

By the use of properties of a parallelogram, we can write

QR ∥ XZ

and QR = XZ

QR ∥ XY

and XY = (1/2 *  QR).

Hence, the midpoint theorem is proven.

The Midpoint theorem can also be proved using other geometrical shapes, especially circles, ellipses, and hyperbola, although these are out of the scope of our syllabus for now. It is indeed an ingenious mathematical property we as mankind discovered, and has without a doubt made our dealing with all objects a much easier task, especially from a mathematical perspective.