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 :
STATEMENT | REASON |
D is the mid-point of AB and F is the mid-point of AC. | [Given] [By construction] [By mid-point theorem] This is not possible that two lines parallel to the same line intersect each other. [DE and DF intersect each other at D] So, our supposition is wrong. |
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.
1. What is the Mid-Point Theorem? |
2. What is the importance of the Mid-Point Theorem? |
3. Can the Mid-Point Theorem be applied to quadrilaterals? |
4. How can the Mid-Point Theorem be proved? |
5. What are some solved problems related to the Mid-Point Theorem? |
|
Explore Courses for Class 9 exam
|