Question

15 multiple choice 1 point
in triangle abc, \\( \angle a \\) is a right angle and \\( m \angle b=45^{\circ} \\). find bc. if your answer
is not an integer, leave it in simplest radical form.
\\( 13 \sqrt{2} \mathrm{ft} \\)
\\( 13 \mathrm{ft} \\)
\\( 26 \mathrm{ft} \\)
\\( 26 \sqrt{2} \mathrm{ft} \\)

Explanation:

Step1: Determine the type of triangle

Since \(\angle A = 90^{\circ}\) and \(\angle B=45^{\circ}\), then \(\angle C=180^{\circ}-\angle A - \angle B=180^{\circ}-90^{\circ}-45^{\circ} = 45^{\circ}\). So, \(\triangle ABC\) is a \(45 - 45-90\) right - triangle. In a \(45 - 45-90\) triangle, the sides are in the ratio \(a:a:a\sqrt{2}\), where \(a\) is the length of the legs and \(a\sqrt{2}\) is the length of the hypotenuse.

Step2: Identify the legs and hypotenuse

In \(\triangle ABC\), \(AC = 13\) ft is one leg. Let \(AB\) be the other leg. Since \(\angle B=\angle C = 45^{\circ}\), \(AB = AC=13\) ft (legs of a \(45 - 45-90\) triangle are equal). Let \(BC\) be the hypotenuse.
Using the Pythagorean theorem \(BC^{2}=AB^{2}+AC^{2}\). Substituting \(AB = 13\) and \(AC = 13\), we get \(BC^{2}=13^{2}+13^{2}=169 + 169=2\times169\). Then \(BC=\sqrt{2\times169}\).
Since \(\sqrt{2\times169}=\sqrt{2}\times\sqrt{169}\) and \(\sqrt{169} = 13\), so \(BC = 13\sqrt{2}\) ft.

Answer:

\(13\sqrt{2}\text{ ft}\)