Mid point theorem candid lab activity class-9
Answers
Mid point theorem
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.
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 || BC
DE = (1/2 * BC).
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)
angle CEF = angle AED {vertically opposite angles}
EF = DE { by construction}
hence,
∆ CFE is congruent to ∆ ADE {by SAS}
Therefore,
angle CFE = angle ADE {by c.p.c.t.}
angle FCE= angle DAE {by c.p.c.t.}
and CF = AD {by c.p.c.t.}
The angles, angle CFE and angleADE are the alternate interior angles. Assume CF and AB as two lines which are intersected by the transversal DF.
In a similar way, angle FCE and angle 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.
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