Mid Point Theorem

Geometry is one among the fundamental and essential branches of mathematics. This field deals with the geometrical problems and figures which are based on the properties. One of the important theorems in the field of geometry that deals with the properties of triangles are called as the Mid- Point Theorem.

The theory of midpoint theorem is used in the coordinate geometry stating that the midpoint of the line segment is an average of the endpoints. To solve an equation by the use of this Theorem, it is important that both the ‘x’ and the ‘y’ coordinates should be known. The Mid- Point Theorem is also useful in the fields of calculus and algebra.

MidPoint Theorem Statement

The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the triangle.”

Mid- Point Theorem

MidPoint Theorem Proof

If midpoints of any of the sides of a triangle are adjoined by the line segment, then the line segment is said to be in parallel to all the remaining sides and also will measure about half of the remaining sides.

Consider the triangle ABC, as shown in the above figure,

Let E and D be the midpoints of the sides AC and AB. Then the line DE is said to be parallel to the sides BC, whereas the side DE is half of the side BC; i.e.

DE is parallel to BC

\(DE \parallel BC\)

DE = (1/2 *  BC).

Now consider the below figure,

Mid- Point Theorem

Construction-  Extend the line segment DE and produce it to F such that, EF=DE.

In the triangle, ADE, and also  the triangle CFE

EC= AE —–   (given)

∠CEF = ∠AED {vertically opposite angles}

EF = DE { by construction}

hence,

△ CFE ≅  △ ADE {by SAS}

Therefore,

∠CFE = ∠ADE {by c.p.c.t.}

∠FCE= ∠DAE    {by c.p.c.t.}

and CF = AD {by c.p.c.t.}

The angles, ∠CFE and ∠ADE are the alternate interior angles. Assume  CF and AB as two lines which are intersected by the transversal DF.

In a similar way, ∠FCE and ∠DAE are the alternate interior angles.  Assume CF and AB are the two lines which are intersected by the transversal AC.

Therefore, CF ∥ AB

So, CF ∥ BD

and CF = BD  {since BD = AD, it is proved that CF = AD}

Thus, BDFC forms a parallelogram.

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

BC ∥ DF

and BC = DF

BC ∥ DE

and DE = (1/2 *  BC).

Hence, the midpoint theorem is  Proved.

MidPoint Theorem Formula

In Coordinate Geometry, midpoint theorem refers to the midpoint of the line segment. It defines the coordinate points of the midpoint of the line segment can be found by taking the average of the coordinates of the given endpoints. The midpoint formula is used to determine the midpoint between the two given points.

If P1(x1, y1) and p2(x2, y2) are the coordinates  two given endpoints, then the midpoint formula is given as:

Midpoint = ((x1+x2)/2, (y1+y2)/2)

The converse of MidPoint Theorem

The converse of the midpoint theorem states that ” if a line is drawn through the midpoint of one side of a triangle, and parallel to the other side, it bisects the third side”.

Midpoint Theorem Example

The example is given below to understand the midpoint theorem.

Example:

In a triangle ABC, the midpoints of BC, CA, AB are D, E, and F respectively. Find the value of EF, if the value of BC = 14 cm

Solution:

Given: BC = 14 cm

To find the value of EF.

Midpoint Theorem Example

If F is the midpoint of AB and E is the midpoint of AC, then we can write it as:

EF = 1/2 (BC)

Now, substitute the value of BC

EF = 1/2(14)

EF = 7 cm

Therefore, the value of EF = 7cm.

The Mid- Point Theorem can also be proved by the use of triangles. The line segment which is on the angle, suppose two lines are drawn in parallel to the x and the y-axis which begin at endpoints and also the midpoint, then the result is said to be two similar triangles. This relation of these triangles forms the Mid- Point Theorem.

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