Question

find the length of \\( \overline { b c } \\).

Explanation:

Step1: Use the sine rule

In a triangle, the sine rule states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let \(BC = a\), \(AC = 4\sqrt{2}\), \(\angle A = 45^{\circ}\), \(\angle B = 90^{\circ}\), \(\angle C = 45^{\circ}\). But wait, no, wait the triangle has angles \(45^{\circ}, 45^{\circ}, 90^{\circ}\)? No, wait no, wait the triangle: \(\angle B = 90^{\circ}\) (since it's a right - angled triangle assumption from the notation), \(\angle A=45^{\circ}\), \(\angle C = 45^{\circ}\). Wait no, no, wait using Pythagoras for a \(45 - 45-90\) triangle: sides are in the ratio \(1:1:\sqrt{2}\). If hypotenuse \(AC = 4\sqrt{2}\), then legs \(AB = BC\). Let \(AB = BC=x\). By Pythagoras \(x^{2}+x^{2}=(4\sqrt{2})^{2}\).

Step2: Solve the Pythagorean equation

\(2x^{2}=32\), then \(x^{2}=16\), so \(x = 4\).

Answer:

\(4\)