Question

determining side lengths of special right triangles. what are the lengths of the other two sides of the triangle? ac = 5\sqrt{3} and bc = 5. ac = 5 and bc = 5\sqrt{5}. ac = 5 and bc = 5\sqrt{3}.

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 right - triangle, if the hypotenuse \(c\) is given, and the sides are in the ratio \(1:\sqrt{3}:2\) (where the side opposite the 30° angle is \(a\), the side opposite the 60° angle is \(b\), and the hypotenuse is \(c\), \(a:x:2x\) with \(c = 2x\)).

Step2: Find the side opposite 30°

Given hypotenuse \(AB = 10\). Let the side opposite the 30° angle (\(AC\)) be \(x\). Since \(c = 2x\) and \(c=10\), then \(2x = 10\), so \(x=\frac{10}{2}=5\).

Step3: Find the side opposite 60°

The side opposite the 60° angle (\(BC\)) is \(\sqrt{3}x\). Substituting \(x = 5\), we get \(BC = 5\sqrt{3}\).

Answer:

AC = 5 and BC = 5√3