Question

solve the right triangle.
write your answers in simplified, rationalized form. do not round.
$ab =$
$m\angle a =$
$m\angle c =$

Explanation:

Step1: Use Pythagorean theorem to find \(AB\)

In right - triangle \(ABC\) with \(AC = 6\sqrt{2}\), \(BC=6\), by \(AC^{2}=AB^{2}+BC^{2}\). Let \(AB = x\), then \((6\sqrt{2})^{2}=x^{2}+6^{2}\).

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Step2: Use trigonometric ratios to find \(\angle A\)

\(\sin C=\frac{AB}{AC}\), \(\sin C=\frac{6}{6\sqrt{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\), so \(\angle C = 45^{\circ}\)
Since the sum of angles in a triangle is \(180^{\circ}\) and \(\angle B = 90^{\circ}\), \(\angle A+\angle C=90^{\circ}\). So \(\angle A = 45^{\circ}\)

Answer:

\(AB = 6\), \(m\angle A=45^{\circ}\), \(m\angle C = 45^{\circ}\)