Question

in the diagram shown of right triangle bac, ( mangle a = 90 ), ( mangle b = 45 ), and ( ac = 8 ). what is the length of ( bc )?

Explanation:

Step1: Determine the type of triangle

Since \(m\angle A = 90^{\circ}\) and \(m\angle B=45^{\circ}\), then \(m\angle C = 180^{\circ}-90^{\circ}-45^{\circ}=45^{\circ}\). So, \(\triangle BAC\) is a \(45 - 45-90\) right - triangle. In a \(45 - 45-90\) triangle, the legs are equal. So \(AB = AC = 8\).

Step2: Use the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a,b\) are the legs. Here \(a = AB = 8\), \(b = AC = 8\), and \(c=BC\). Then \(BC^{2}=AB^{2}+AC^{2}\). Substitute \(AB = 8\) and \(AC = 8\) into the formula: \(BC^{2}=8^{2}+8^{2}=64 + 64=128\).

Step3: Solve for \(BC\)

Take the square root of both sides: \(BC=\sqrt{128}=\sqrt{64\times2}=8\sqrt{2}\).

Answer:

\(8\sqrt{2}\)