Question

what is the length of side ab? do not round your answer.
type your answer in the box.
______ units

Explanation:

Step1: Identify the triangle type

The triangle \( \triangle ABC \) is a right - triangle with \( \angle A = 90^{\circ} \), \( \angle C=30^{\circ} \) and \( AC = 6 \). We can use trigonometric ratios to find the length of \( AB \). Recall that in a right - triangle, \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). For \( \angle C = 30^{\circ} \), the opposite side to \( \angle C \) is \( AB \) and the adjacent side is \( AC \). So \( \tan(30^{\circ})=\frac{AB}{AC} \).

Step2: Substitute the known values

We know that \( AC = 6 \) and \( \tan(30^{\circ})=\frac{1}{\sqrt{3}} \). Substituting these values into the formula \( \tan(30^{\circ})=\frac{AB}{AC} \), we get \( \frac{1}{\sqrt{3}}=\frac{AB}{6} \).

Step3: Solve for \( AB \)

To solve for \( AB \), we can cross - multiply. So \( AB=\frac{6}{\sqrt{3}} \). We can rationalize the denominator by multiplying the numerator and denominator by \( \sqrt{3} \). \( AB = \frac{6\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{6\sqrt{3}}{3}=2\sqrt{3} \). We can also use the relationship in a 30 - 60 - 90 triangle. In a 30 - 60 - 90 triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \) (opposite to 30°, 60°, 90° respectively). If the side adjacent to 30° (which is the longer leg, \( AC \)) has length \( 6 \), and the shorter leg ( \( AB \)) is related to the longer leg by the ratio \( \frac{\text{shorter leg}}{\text{longer leg}}=\frac{1}{\sqrt{3}} \), so \( AB=\frac{6}{\sqrt{3}} = 2\sqrt{3} \) (after rationalizing) or we can also think of it as using \( \tan(30^{\circ})=\frac{AB}{6} \), so \( AB = 6\times\tan(30^{\circ})=6\times\frac{\sqrt{3}}{3}=2\sqrt{3} \). Another way is to use the tangent ratio or the properties of 30 - 60 - 90 triangles. Alternatively, we can use the cotangent function. \( \cot(30^{\circ})=\sqrt{3}=\frac{AC}{AB} \), so \( AB=\frac{AC}{\cot(30^{\circ})}=\frac{6}{\sqrt{3}} = 2\sqrt{3} \) (after rationalizing).

Answer:

\( 2\sqrt{3} \)