Question

explain mid-point theorem​

Answer 1

≡QUESTION≡

Explain the mid-point theorem.

║⊕ANSWER⊕║

Objective:

To verify the mid-point theorem for a triangle.

Theorem :

The line segment connecting the midpoints of  two sides of a triangle is parallel to the third side and is congruent to one half of the third side.

Given in the figure A :  AP=PB

                                       AQ=QC.

To prove: PQ || BC and PQ=1/2 BC

▲ APQ ≅ ▲ QRC

AQ=QC

[midpoint]

∠ APQ = ∠QRC

[Corresponding angles for parallel lines cut by transversal].

∠PBR=∠QRC=∠APQ

[Corresponding angles for parallel lines cut by an transversal].

∠RQC=∠PAQ

[When 2 pairs of corresponding angles are congruent in a triangle, the third pair is also congruent.]

∴ ▲APQ ≅ ▲QRC

AP=QR=PB and PQ=BR=RC.

Since midpoints are unique, and the lines connecting points are unique, the proposition is proven.

Answer 2

hi mate,

solution: 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.”

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 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 side BC, whereas the side DE is half of the side BC; i.e.

DE∥BC

DE = (1/2 * BC).

I hope it helps you.