Question

11 multiple choice 1 point in triangle abc, ∠a is a right angle and m∠b = 45°. find bc. if your answer is not an integer, leave it in simplest radical form. not drawn to scale 30 ft 30√2 ft 15√2 ft 15 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 legs are equal. Let \(AB = AC = 15\) ft.

Step2: Use the Pythagorean theorem

The Pythagorean theorem states that \(BC^{2}=AB^{2}+AC^{2}\). Substitute \(AB = 15\) and \(AC = 15\) into the formula: \(BC^{2}=15^{2}+15^{2}\).

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Another way: In a \(45 - 45-90\) triangle, the hypotenuse \(c\) and the leg \(a\) are related by the formula \(c = a\sqrt{2}\). Here \(a = 15\) (either \(AB\) or \(AC\)), so \(BC=15\sqrt{2}\) ft.

Answer:

C. \(15\sqrt{2}\) ft