Mid Point Theorem

Some solved problems on Mid Point -Theorem

1) In a parallelogram ABCD, E and F are the mid-points 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

Statements	Reasons
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)

AF and EC trisects the diagonal BD.

_________________________________________________________________
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.

Statements	Reasons
1) PQ || BC	1) Given
2) ∠AQP = ∠ABC	2) Corresponding angles
3) ∠AQP = 900	3) Since ∠ABC =900
4) ∠AQP + ∠BQP = 1800	4) Linear pair angles
5) ∠AQP = ∠BQP = 900	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

PA = PB = 1/2 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

Home Page