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
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| 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.
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| 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 |
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