What's Mid-point Theorem ??
Answers
Answer:
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
Anytime you have a line segment that connects two sides of a triangle at the midpoints, you automatically know that the sides are cut in half, and that the segment is parallel to the third side of the triangle. Parallel sides are shown by using this symbol ||. You also know the line segment is one-half the length of the third side.
✨Hope it will help you.✨
Midpoint 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. Let us observe the figure given above. We can see the ΔABC,
Point D and E are the midpoints of side AB and side AC respectively. Also, the segment DE connects the two sides at the midpoints, then DE || BC and DE is half the length of side BC.
If we know the length of BC then it is very convenient to find the length of DE as DE is half of BC. It also allows us to find the length of sides AE, EC, BD, and DA. Since DE is parallel to BC we know that the distance between these two line segment is equal.
Also, ∠ADE = ∠ABC
So, DE || BC
Proof of the Theorem
Given: In triangle ABC, D and E are midpoints of AB and AC respectively.
To Prove:
DE || BC
DE = 1/2 BC
Construction: Draw CR || BA to meet DE produced at R. (Refer the above figure)
∠EAD = ∠ECR. (Pair of alternate angles) ———- (1)
AE = EC. (∵ E is the mid-point of side AC) ———- (2)
∠AEP = ∠CQR (Vertically opposite angles) ———- (3)
Thus, ΔADE ≅ ΔCRE (ASA Congruence rule)
DE = 1/2 DR ———- (4)
But, AD= BD. (∵ D is the mid-point of the side AB)
Also. BD || CR. (by construction)
In quadrilateral BCRD, BD = CR and BD || CR
Therefore, quadrilateral BCRD is a parallelogram.
BC || DR or, BC || DE
Also, DR = BC (∵ BCRD is a parallelogram)
⇒ 1/2 DR = 1/2 BC
The Converse of MidPoint Theorem
The line drawn through the mid–point of one side of a triangle and parallel to another side bisects the third side.
