OVERVIEW

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

The theory of MIDPOINT theorem is used in coordinate geometry stating that the midpoint of the line segment is an average of the endpoints. Both the ‘x’ and the ‘y’ coordinates must be known for solving an equation using this theorem. 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 length of the third side.”

MidPoint Theorem Proof

If the line segment adjoins midpoints of any of the sides of a triangle, then the line segment is said to be parallel to all the remaining sides, and it measures 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 side BC, WHEREAS the side DE is half of the side BC; i.e.

DE∥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 triangle ADE and CFE,

EC = AE —– (given)

∠CEF = ∠AED (vertically opposite angles)

EF = DE (by construction)

By SAS congruence criterion,

△ CFE ≅ △ ADE

Therefore,

∠CFE = ∠ADE {by c.p.c.t.}

∠FCE= ∠DAE {by c.p.c.t.}

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

∠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 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

⚕️ NONEXISTENT⚕️