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.
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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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