Mid Point Theorem

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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.

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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.


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

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