state mid point theorem and converse of mid point theorem.
Answers
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.”
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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 a triangle ABC,
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).
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)
∠DAE = ∠ECF (alternate angles)
By ASA 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.
MidPoint Theorem Formula
If P1(x1, y1) and P2(x2, y2) are the coordinates of two given endpoints, then the midpoint formula is given as:
Midpoint = [(x1 + x2)/2, (y1 + y2)/2]
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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”.
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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.”
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”
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