7.6: Method of coordinates
The following exercise is important; it shows that our axiomatic definition agrees with the model definition.
Let \(\ell\) and \(m\) be perpendicular lines in the Euclidean plane. Given a point \(P\), let \(P_{\ell}\) and \(P_m\) denote the foot points of \(P\) on \(\ell\) and \(m\) respectively.
- Show that for any \(X \in \ell\) and \(Y \in m\) there is a unique point \(P\) such that \(P_{\ell} = X\) and \(P_m = Y\).
- Show that \(PQ^2 = P_{\ell}Q_{\ell}^2 + P_m Q_m^2\) for any pair of points \(P\) and \(Q\).
- Conclude that the plane is isometric to \((\mathbb{R}^2, d_2)\).
- Hint
-
(a). Use the uniqueness of the parallel line ( Theorem 7.1.1 ).
(b). Use Lemma 7.5.1 and the Pythagorean theorem ( Theorem 6.2.1 )
Once this exercise is solved, we can apply the method of coordinates to solve any problem in Euclidean plane geometry. This method is powerful and universal; it will be developed further in Chapter 18.
Use the Exercise \(\PageIndex{1}\) to give an alternative proof of Theorem 3.5.1 in the Euclidean plane.
That is, prove that given the real numbers \(a, b\), and \(c\) such that
\(0 < a \le b \le c \le a + b\),
there is a triangle \(ABC\) such that \(a = BC\), \(b = CA\), and \(c = AB\).
- Hint
-
Set \(A = (0, 0), B = (c, 0)\), and \(C = (x, y)\). Clearly, \(AB = c\), \(AC^2 = x^2 + y^2\) and \(BC^2 = (c - x)^2 + y^2\).
It remains to show that there is a pair of real numbers \((x, y)\) that satisfy the following system of equations:
\(\begin{cases} b^2 = x^2 + y^2 \\ a^2 = (c- x)^2 + y^2 \end{cases}\)
if \(0 < a \le b \le c \le a + c\).
Consider two distinct points \(A = (x_A, y_A)\) and \(B = (x_B, y_B)\) on the coordinate plane. Show that the perpendicular bisector to \([AB]\) is described by the equation
\(2 \cdot (x_B - x_A) \cdot x + 2 \cdot (y_B - y_A) \cdot y = x_B^2 + y_B^2 - x_A^2 - y_B^2\).
Conclude that line can be defined as a subset of the coordinate plane of the following type:
- Solutions of an equation \(a \cdot x + b \cdot y = c\) for some constants \(a, b\), and \(c\) such that \(a \ne 0\) or \(b \ne 0\).
- The set of points \((a \cdot t + c, b \cdot t + d)\) for some constants \(a, b, c\), and \(d\) such that \(a \ne 0\) or \(b \ne 0\) and all \(t \in \mathbb{R}\).
- Hint
-
Note that \(MA = MB\) if and only if
\((x - x_A)^2 + (y - y_A)^2 = (x - x_B)^2 + (y - y_B)^2\)
where \(M = (x, y)\). To prove the first part, simplify this equation. For the remaining parts use that any line is a perpendicular bisector to some line segment.