Question

fill in the blank 1 point if b=$5sqrt{3}$, find a and c. a= choose your answer... c= choose your answer...

Explanation:

Step1: Identify the triangle type

This is a 30 - 60 - 90 right - triangle. In a 30 - 60 - 90 triangle, the ratios of the sides are \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest side (let's call it \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

In the given triangle, side \(b\) is opposite \(60^{\circ}\) (since the right angle is between \(a\) and \(b\), angle at the bottom is \(30^{\circ}\), angle at the top is \(60^{\circ}\)), side \(a\) is opposite \(30^{\circ}\), and \(c\) is the hypotenuse.

We know that the side opposite \(60^{\circ}\) (which is \(b\)) is \(x\sqrt{3}\), and we are given \(b = 5\sqrt{3}\). Let the side opposite \(30^{\circ}\) (which is \(a\)) be \(x\). Then, since the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), we have \(x\sqrt{3}=5\sqrt{3}\).

Step2: Solve for \(a\) (side opposite \(30^{\circ}\))

From \(x\sqrt{3}=5\sqrt{3}\), we can divide both sides by \(\sqrt{3}\) to get \(x = 5\). So, \(a=x = 5\) (because \(a\) is opposite \(30^{\circ}\)).

Step3: Solve for \(c\) (hypotenuse)

The hypotenuse \(c\) in a 30 - 60 - 90 triangle is \(2x\). Since \(x = 5\), then \(c = 2x=2\times5 = 10\).

Answer:

\(a = 5\), \(c = 10\)