1.6: Half-lines and segments

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Assume there is a line \(l\) passing thru two distinct points \(P\) and \(Q\). In this case we might denote \(l\) as \((PQ)\). There might be more than one line thru \(P\) and \(Q\), but if we write \((PQ)\) we assume that we made a choice of such line.

We will denote by \([PQ)\) the half-line that starts at \(P\) and contains \(Q\). Formally speaking, \([PQ)\) is a subset of \((PQ)\) which corresponds to \([0,\infty)\) under an isometry \(f: (PQ) \to \mathbb{R}\) such that \(f(P) = 0\) and \(f(Q) > 0\).

The subset of line \((PQ)\) between \(P\) and \(Q\) is called the segment between \(P\) and \(Q\) and denoted by \([PQ]\). Formally, the segment can be defined as the intersection of two half-lines: \([PQ] = [PQ) \cap [QP)\).

Exercise \(\PageIndex{1}\)

Show that

(a) if \(X \in [PQ)\), then \(QX = |PX - PQ|\);

(b) if \(X \in [PQ]\), then \(QX + XQ = PQ\).

Hint

Fix an isometry \(f: (PQ) \to \mathbb{R}\) such that \(f(P) = 0\) and \(f(Q) = q > 0\).

Assume that \(f(X) = x\). By the definition of the half-line \(X \in [PQ)\) if and only if \(x \ge 0\). Show that the latter holds if and only if \(|x - q| = ||x| - |q||\). Hence (a) follows.

To prove (b), observe that \(X \in [PQ]\) if and only if \(0 \le x \le q\). Show that the latter holds if and only if \(|x - q| + |x| = |q|\).