State and Prove Midpoint Theorem Class 9
Midpoint Theorem Statement: “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”.
Given: A △ABC in which D and E are the mid-points of sides AB and AC respectively. DE is joined to F.
To Prove: DE || BC and DE = ½ x BC
Construction: Produce the line segment DE to F, such that DE = EF and Join FC.
Proof: In △s AED and CEF, we have
⇒ AE = CE [E is the midpoint of AC]
⇒ ∠AED = ∠CEF [Vertically opposite angles]
and, DE = EF [by construction]
So, by SAS criterion of congruence, we have
⇒ AED ⩭ ACEF
⇒ AD = CF …………….(a)
and, ∠ADE = ∠CFE [c.p.c.t.] …………. (b)
Now, D is the midpoint of AB
⇒ AD = DB
⇒ DB = CF [From (i) AD= CF] …………. (c)
⇒ ∠ADE = ∠CFE
i.e., alternate interior angles are equal
⇒ AD∥FC
⇒ DB∥FC …………. (d)
From eq. (c) and (d), we find that DBCF is a quadrilateral, in which
⇒ DF∥BC and DF = BC
But, D, E, and F are collinear and DE=EF
⇒ ∴ DF∥BC and DE = ½ x BC
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