Mid-Point Theorem Class 9 - Quadrilaterals, CBSE Mathematics PDF Download

MID- POINT THEOREM

THEOREM 1 : The line segment joining the mid-points of two sides of a triangle is parallel to the third side.
GIVEN : E and F are the mid-points of the sides AB and AC respectively of the ΔABC.

TO PROVE : EF || BC.
CONSTRUCTION : Throught the vertex C, CG is drawn parallel to AB and it meets EF (produced) in G.
PROOF :

Converse of Theorem 1 : The line drawn through the mid-point of one side of a triangle and parallel to another side of the triangle.
GIVEN : ABC is a triangle in which D is mid-point of AB and DE || BC.

TO PROVE : E is the mid-point of AC.
CONSTRUCTION : Let E is not the mid-point of AC. If possible, let F is the mid-point of AC. Join DF.
PROOF :

Hence proved.

THEOREM 2 : Length of the line segment joining the mid points of two sides of a triangle is equal to half the length of the third side.

GIVEN : In ΔABC, EF is the line segement joining the mid-points of the sides AB and AC of ΔABC.
TO PROVE :

EF =

CONSTRUCTION : Through C, draw CG || BA. CG meets EF (produced) at G.
PROOF :

Ex. In the following figure, D, E and F are respectively the mid-points of sides BC, CA and AB of an equilateral triangle ABC. Prove that ΔDEF is also an equilateral triangle.

Sol. Given : D, E and F are respectively the mid-points of sides BC, CA and AB of an equilateral triangle ABC.
To prove : ΔDEF is also an equilateral triangle.
Proof : since the segment joining the mid points of two sides of a triangle is half of the third side. Therefore D and E are the mid point of BC and AC respectively.

E and F are the mid point of AC and AB respectively

F and D are the mid point of AB and BC respectively

Ex. In figure, D and E are the mid-point of the sides AB and AC respectively of ΔABC. If BC = 5.6 cm, find DE.

Sol. D is mid-point of AB and E is mid-point of AC.

Ex. In figure, E and F are mid-points of the sides AB and AC respectively of the ABC, G and H are mid-points of the sides AE and AF respectively of the AEF. If GH = 1.8 cm, find BC.