Question

4 fill in the blank 1 point if c=18, find a and b. a = choose your answer... b = choose your answer...

Explanation:

Step1: Identify triangle type

This is a 30-60-90 right triangle. In such a triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite \(30^\circ\) is the shortest (let's call it \(a\) here? Wait, no: wait, angle \(30^\circ\) is at the bottom right, so the side opposite \(30^\circ\) is \(a\) (the vertical leg), the side opposite \(60^\circ\) is \(b\) (the horizontal leg), and hypotenuse \(c = 18\). Wait, ratio: opposite \(30^\circ\): \(a\), opposite \(60^\circ\): \(b\), hypotenuse \(c\). So ratio \(a : b : c = 1 : \sqrt{3} : 2\). So \(c = 2x\), \(a = x\), \(b = x\sqrt{3}\).

Step2: Find \(a\)

Given \(c = 18 = 2x\), solve for \(x\): \(x=\frac{18}{2}=9\). So \(a = x = 9\).

Step3: Find \(b\)

Using \(b = x\sqrt{3}\), and \(x = 9\), so \(b = 9\sqrt{3}\) (or approximately \(15.588\), but exact form is \(9\sqrt{3}\)).

Answer:

\(a = 9\), \(b = 9\sqrt{3}\) (or \(b\approx15.59\) if decimal is preferred, but exact is \(9\sqrt{3}\))