Question

in triangle abc, the measure of angle a is 30° and the measure of angle b is 90°. if the length of side bc is 42 centimeters, what is the length, in centimeters, of side ab? a 42 b 42√2 c 84 d 42√3

Explanation:

Step1: Identify Triangle Type

Triangle \(ABC\) is a right - triangle with \(\angle B = 90^{\circ}\), \(\angle A=30^{\circ}\), so \(\angle C = 60^{\circ}\). Side \(BC\) is opposite \(\angle A\) (opposite \(30^{\circ}\) angle), side \(AB\) is adjacent to \(\angle A\), and side \(AC\) is the hypotenuse.

Step2: Recall Trigonometric Ratios

In a right - triangle, \(\tan A=\frac{\text{opposite}}{\text{adjacent}}\). We know that \(\tan30^{\circ}=\frac{BC}{AB}\), and \(BC = 42\) cm, \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\).

Step3: Solve for \(AB\)

From \(\tan A=\frac{BC}{AB}\), we can re - arrange the formula to \(AB=\frac{BC}{\tan A}\). Substituting \(BC = 42\) and \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\), we get \(AB=\frac{42}{\frac{1}{\sqrt{3}}}=42\sqrt{3}\) cm. We can also use the properties of 30 - 60 - 90 triangles. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (opposite \(30^{\circ}\), opposite \(60^{\circ}\), hypotenuse). The side opposite \(30^{\circ}\) ( \(BC\)) is \(x\), the side opposite \(60^{\circ}\) ( \(AB\)) is \(x\sqrt{3}\), and the hypotenuse is \(2x\). Here \(x = 42\), so the side opposite \(60^{\circ}\) ( \(AB\)) is \(42\sqrt{3}\).

Answer:

D. \(42\sqrt{3}\)