Parallel sides and quadrilaterals

Parallel sides and quadrilaterals

Two lines \(AB\) and \(DC\) that are not the same line are parallel when \(\overrightarrow{AB}=k\overrightarrow{DC}\) for some number \(k\neq0\). The value of \(k\) also compares their lengths: \(AB=|k|\,DC\). For a quadrilateral \(ABCD\):

1. If \(\overrightarrow{AB}=\overrightarrow{DC}\), the sides \(AB\) and \(DC\) are parallel and equal in length, so \(ABCD\) is a parallelogram.
2. If \(\overrightarrow{AB}=k\overrightarrow{DC}\) with \(k>0\) and \(k\neq1\), the sides are parallel but of different lengths, so \(ABCD\) is a trapezium with \(AB\) parallel to \(DC\).

Method

1. Express the two side vectors in the same given vectors.
2. Decide whether one is a number times the other, and find that number.
3. Write the conclusion in words: which sides are parallel, how their lengths compare, and the name of the quadrilateral.

Example (left figure): \(\overrightarrow{AB}=7\mathbf{p}\) and \(\overrightarrow{DC}=4\mathbf{p}\), so \(\overrightarrow{AB}=\dfrac74\overrightarrow{DC}\). Hence \(AB\) is parallel to \(DC\) and \(\dfrac74\) times as long, so \(ABCD\) is a trapezium.

Points one third of the way along two sides

In triangle \(EFG\) (right figure) let \(H\) and \(J\) be the points one third of the way from \(E\) along \(EF\) and \(EG\), so \(\overrightarrow{EH}=\dfrac13\overrightarrow{EF}\) and \(\overrightarrow{EJ}=\dfrac13\overrightarrow{EG}\). Then \(\overrightarrow{HJ}=\overrightarrow{EJ}-\overrightarrow{EH}=\dfrac13(\overrightarrow{EG}-\overrightarrow{EF})=\dfrac13\overrightarrow{FG}\). So \(HJ\) is parallel to \(FG\) and one third as long.

Check

Give the reason with the conclusion, for example: "Since \(\overrightarrow{AB}=\dfrac74\overrightarrow{DC}\), \(AB\) is parallel to \(DC\) and \(AB=\dfrac74DC\), so \(ABCD\) is a trapezium." Parallel sides alone do not make a parallelogram: the lengths must be equal too.