Geometry Challenge

[Thomas Klemm]

(01-09-2024 12:27 PM)Albert Chan Wrote:  How does this work?
Why the subtraction and division?

Consider \(B\) the origin of \(\mathbb{C}\).

Then:

\(
\begin{align}
A &= (1 \measuredangle 70^{\circ}) \\
\\
C &= 1 \\
\\
D
&= C + (1 \measuredangle 10^{\circ}) \\
&= 1 + (1 \measuredangle 10^{\circ}) \\
\\
\overline{AD}
&= D - A \\
&= 1 + (1 \measuredangle 10^{\circ}) - (1 \measuredangle 70^{\circ}) \\
\\
\overline{AB}
&= B - A \\
&= -A \\
\end{align}
\)

To get the angle between \(\overline{AD}\) and \(\overline{AB}\) we can divide them:

\(
\begin{align}
\frac{\overline{AD}}{\overline{AB}} = \frac{1 + (1 \measuredangle 10^{\circ}) - (1 \measuredangle 70^{\circ})}{- (1 \measuredangle 70^{\circ})}
\end{align}
\)

Since \(\left \| \overline{AB} \right \| = 1\) we also get \(\left \| \overline{AD} \right \|\).