The line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.

Converse of mid-point theorem

The line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.

Given:A triangle ABC,E is mid-point of AB. Line l drawn through E and parallel to BC meeting AC at F.

To prove. AF=FC

Construction: Through C, draw a line m parallel BA to meet line l at D.

Proof:

▆

The line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.

Given:A triangle ABC,E is mid-point of AB. Line l drawn through E and parallel to BC meeting AC at F.

To prove. AF=FC

Construction: Through C, draw a line m parallel BA to meet line l at D.

Proof:

Statements |
Reasons |

1. EBCD is a parallelogram |
1. EF∥BC (given), BA∥CD (const.). |

2. BE=CD |
2. Opp.sides of a parallelogram are equal. |

3. EA=BE |
3. E is mid-point of AB(given). |

4. EA=CD |
4. From 2 and 3. |

In △AEF and △CDF | |

5. ∠EAF=∠DCF |
5. Alt. ∠s, CD ∥BA and AC is a transversal. |

6. ∠EFA=∠DFC | 6. Vert.opp.∠s. |

7. EA=CD |
7. From 4 |

8. △AEF≌△CDF | 8. AAS rule of ccongruency |

9. AF=FC |
9. c.p.c.t. |

Hence, F is mid-point of AC. |