Mid Point Theorem
Some solved problems on Mid Point Theorem
1) In a parallelogram ABCD, E and F are the midpoints of sides AB and CD respectively. Prove that the line segments AF and EC trisect the diagonal BD.
Given : ABCD is a parallelogram. E and F are mid points.
Prove that : AF and EC trisects the diagonal BD i.e BP = PQ = QD



1) E and F are mid points of AB and CD respectively  1) Given  
2) AE = 1/2 AB and CF = 1/2 CD  2) Definition of mid point  
3) ABCD is a parallelogram  3) Given  
4)AB =CD and AB  CD 
4) Properties of parallelogram  
5) AE = FC and AE  FC  5) From (2) and (4)  
6) AECF is a parallelogram  6) Properties of parallelogram and from (5)  
7) FA  CE and FQ  CP  7) Properties of Parallelogram  
8) F is the mid point of CD and FQ CP  8) By Mid Point Theorem  
9) Q is the mid point of DP ⇒PQ = QD  9) By Mid Point Theorem  
10)Similarly, P is the mid point of BQ ⇒ BP = PQ  10) By mid point theorem  
11) BP= PQ = QD  11) From (9) and (10)  
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2) In ΔABC , right angled at B; and P is the mid point of AC. Prove that 1) PQ ⊥ AB 2) Q is the mid point of AB 3) PB = PA = ½ AC
Given : ΔABC right angled at B
P is the mid point of AC
Prove that : 1) 1) PQ ⊥ AB 2) Q is the mid point of AB 3) PB = PA = ½ AC
Construction : Through P draw PQ  BC meeting AB at Q.



1) PQ  BC  1) Given  
2) ∠AQP = ∠ABC  2) Corresponding angles  
3) ∠AQP = 90^{0}  3) Since ∠ABC =90^{0}  
4) ∠AQP + ∠BQP = 180^{0}  4) Linear pair angles  
5) ∠AQP = ∠BQP = 90^{0}  5) ∠AQP = 90 and from (4)  
6) PQ ⊥ AB  6) From (5)  
7) P is the mid point of AC and PQ BC  7) Given  
8) Q is the mid point. AQ =BQ  8) By mid point theorem and definition of mid point.  
9) ∠AQP = ∠BQP  9) From (5)  
10) PQ =PQ  10) Reflexive (common)  
11) ΔAPQ = ΔBPQ  11) SAS Postulate  
12) PA = PB  12) CPCTC  
13) PA = 1/2 AC  13) Since P is the mid point of AC  
Quadrilateral
• Introduction to Quadrilateral
• Types of Quadrilateral
• Properties of Quadrilateral
• Parallelogram and its Theorems
• Rectangle and its Theorems
• Square and its Theorems
• Rhombus and its Theorems
• Trapezoid (Trapezium)and its Theorems
• Kite and its Theorems
• Mid Point Theorem