Question

lesson = trigonometry with pythagoras theorem question 8 of 10
in the above figure \\( \overline { a b } = \overline { b c } \\), find
the length of \\( \overline { d c } \\).

Explanation:

Step1: Analyze triangle ABD

In right - triangle \(ABD\), \(\angle A = 30^{\circ}\), \(\angle ABD=90^{\circ}\), and \(BD = 8\).
We know that \(\cos A=\frac{AB}{AD}\) and \(\sin A=\frac{BD}{AD}\). Using \(\sin A=\frac{BD}{AD}\), since \(\sin30^{\circ}=\frac{1}{2}\) and \(BD = 8\), we have \(\frac{1}{2}=\frac{8}{AD}\), so \(AD = 16\). Also, using \(\cos30^{\circ}=\frac{\sqrt{3}}{2}=\frac{AB}{AD}\), \(AB=\frac{\sqrt{3}}{2}\times16 = 8\sqrt{3}\).

Step2: Use the Pythagorean theorem in triangle BDC

Since \(AB = BC\) (given), \(BC=8\sqrt{3}\). In right - triangle \(BDC\), by the Pythagorean theorem \(DC^{2}=BD^{2}+BC^{2}\).
Substitute \(BD = 8\) and \(BC = 8\sqrt{3}\) into the formula:
\(DC^{2}=8^{2}+(8\sqrt{3})^{2}=64 + 192\).
\(DC^{2}=256\).

Step3: Solve for DC

Take the square root of both sides: \(DC=\sqrt{256}\).

Answer:

\(16\)