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In this section we will discuss GeometryProof 2 based on mid point theorem in triangle. If the line joins the mid point of any two sides of a triangle then it is parallel to third side and it one half of the third side. Given : ΔABC, P is the mid point of AB and PQ ||BC Prove that : Q is the mid point of AC. |
1) P is the mid point | 1) Given (ΔABC) |
2) AP = PB | 2) Definition of mid point |
3) AP / PB = 1 | 3) Ratio of two equal side is 1. |
4) PQ || BC | 4) Given (ΔABC) |
5) ( AP / PB ) = ( AQ / QC ) | 5) Basic Proportionality Theorem |
6) AQ / QC = 1 | 6) From (3) and (5) (Transitivity) |
7) AQ = QC | 7) Cross multiply |
8) Q is the mid point of AC | 8) Definition of mid point |
Given : 1) ABCD is a trapezoid. 2) AB || DC Prove that : OE || DC. Construction : Draw OE such that OE ||DC |
1) OE || DC | 1) Construction (ΔABC) |
2) (BO / OD) = ( BE / EC) | 2) Basic Proportionality Theorem |
3) AB || DC | 3) Given |
4) OE || AB | 4) From (1) and (3) (Transitivity) |
5) (AO / CO) = (BE / EC) | 5) Basic Proportionality Theorem |
6) (AO / CO) = (BO / DO) | 6) From (2) and (5) ( Transitivity) |
7) (AO / BO) = (CO / DO) | 7) From above ( Alternendo) |
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