state and prove equal intercept theorem
Answers
Answer:
The theorem states if a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts. It means that given any three mutually perpendicular lines, a line passing through them forms intercepts in the corresponding ratio of the distances between the lines.
For example, Suppose there are three lines, l, m and n. Keep the distance between l–m twice than the distance between m–n. So any line passing through them, the intercept made by l-m on the line is twice the intercept made by m-n.
Hope it's helpful to you.
Answer:
The theorem states if a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.
Also, ∠ADE = ∠ABC
So, DE || BC
Proof of the Theorem
Intercept
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.