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.

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) |

_________________________________________________________________

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

P is the mid point of AC

Statements |
Reasons |

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 |

• 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

GMAT

GRE

1st Grade

2nd Grade

3rd Grade

4th Grade

5th Grade

6th Grade

7th grade math

8th grade math

9th grade math

10th grade math

11th grade math

12th grade math

Precalculus

Worksheets

Chapter wise Test

MCQ's

Math Dictionary

Graph Dictionary

Multiplicative tables

Math Teasers

NTSE

Chinese Numbers

CBSE Sample Papers