Explain Mid point Theorem, With example .
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Step-by-step explanation:
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 the 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.
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.”
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.
Stay safe and blessed ❤
Answer:
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