Understanding the midpoint theorem to understand the above question. click on the link to Watch the VIDEO explanation: Watch Video
Midpoint Theorem
Let us learn about the midpoint theorem
The theorem states that the straight line joining the midpoints of two sides of a triangle is parallel to and equal to half of the third side.
We are given a triangle ABC with sides AB and AC having P and Q as the midpoints
It is required to prove that a PQ is parallel to BC and PQ is equal to half of BC.
This requires a simple construction
Draw CR parallel BA to meet PQ produced, at R.
Let us compare triangle APQ and triangle CQR
Since BA is parallel to CR alternate angles angle A is equal to angle QCR
Vertically opposite angles angle AQP and angle CQR are equal
AQ is equal to CQ. This data is given
BY AAS congruency triangle APQ is congruent to triangle CQR
Hence, PQ is equal to QR and AP is equal to CR but AP is equal to BP from the given data
Therefore, BP is equal to CR. By construction BP is parallel to CR
Therefore, BCRP is a parallelogram.
PQ is parallel to BC. PQ is half of PR. Since PQ is equal to QR.
But PR is equal to BC since BCRP is a parallelogram. Therefore PQ is equal to half of BC.
Thus, the theorem is proved.
Now let us see a converse of the midpoint theorem
Its states that the straight line draw through the midpoint of one side of a triangle parallel to another, bisects the third side.
We are given a triangle ABC, in which P is the midpoint of AB and PQ is parallel to BC.
To prove that AQ is equal to CQ
Draw CR parallel BA to meet PQ produced, at R.
Proof of the theorem
We are given PR is parallel to BC through construction BP is parallel to CR. Therefore BCRP is a parallelogram. Therefore BP is equal to CR Since the opposite sides of a parallelogram are equal.
But we are given BP is equal to AP. Therefore AP is equal to CR.
Comparing triangle APQ and triangle CQR we have angle A is equal to angle QCR Since they are alternate angles as BP is parallel to CR.
angle AQP is equal to angle CQR since they are vertically opposite angles.
Therefore, by AAS congruency triangle APQ is congruent to triangle CQR
Therefore AQ is equal to CQ.
Thus the converse of the theorem also proved.
Midpoint Theorem
Let us learn about the midpoint theorem
The theorem states that the straight line joining the midpoints of two sides of a triangle is parallel to and equal to half of the third side.
We are given a triangle ABC with sides AB and AC having P and Q as the midpoints
It is required to prove that a PQ is parallel to BC and PQ is equal to half of BC.
This requires a simple construction
Draw CR parallel BA to meet PQ produced, at R.
Let us compare triangle APQ and triangle CQR
Since BA is parallel to CR alternate angles angle A is equal to angle QCR
Vertically opposite angles angle AQP and angle CQR are equal
AQ is equal to CQ. This data is given
BY AAS congruency triangle APQ is congruent to triangle CQR
Hence, PQ is equal to QR and AP is equal to CR but AP is equal to BP from the given data
Therefore, BP is equal to CR. By construction BP is parallel to CR
Therefore, BCRP is a parallelogram.
PQ is parallel to BC. PQ is half of PR. Since PQ is equal to QR.
But PR is equal to BC since BCRP is a parallelogram. Therefore PQ is equal to half of BC.
Thus, the theorem is proved.
Now let us see a converse of the midpoint theorem
Its states that the straight line draw through the midpoint of one side of a triangle parallel to another, bisects the third side.
We are given a triangle ABC, in which P is the midpoint of AB and PQ is parallel to BC.
To prove that AQ is equal to CQ
Draw CR parallel BA to meet PQ produced, at R.
Proof of the theorem
We are given PR is parallel to BC through construction BP is parallel to CR. Therefore BCRP is a parallelogram. Therefore BP is equal to CR Since the opposite sides of a parallelogram are equal.
But we are given BP is equal to AP. Therefore AP is equal to CR.
Comparing triangle APQ and triangle CQR we have angle A is equal to angle QCR Since they are alternate angles as BP is parallel to CR.
angle AQP is equal to angle CQR since they are vertically opposite angles.
Therefore, by AAS congruency triangle APQ is congruent to triangle CQR
Therefore AQ is equal to CQ.
Thus the converse of the theorem also proved.
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