Question

here is triangle abc.
find the length of each leg.
type your answers in the boxes.
ab = units
bc = units

Explanation:

Step1: Identify triangle type

Triangle \(ABC\) is a right - angled triangle at \(B\), with \(\angle A = 60^{\circ}\), \(\angle C=30^{\circ}\) and hypotenuse \(AC = 10\) units. In a \(30 - 60-90\) right - angled triangle, the side opposite the \(30^{\circ}\) angle is half the hypotenuse, and the side opposite the \(60^{\circ}\) angle is \(\frac{\sqrt{3}}{2}\) times the hypotenuse.

Step2: Find length of \(AB\)

The angle at \(C\) is \(30^{\circ}\), and the side opposite to \(\angle C\) is \(AB\). So, by the property of \(30 - 60-90\) triangle, \(AB=\frac{1}{2}\times AC\). Since \(AC = 10\), we have \(AB=\frac{1}{2}\times10 = 5\) units.

Step3: Find length of \(BC\)

The angle at \(A\) is \(60^{\circ}\), and the side opposite to \(\angle A\) is \(BC\). By the property of \(30 - 60-90\) triangle, \(BC=\frac{\sqrt{3}}{2}\times AC\). Since \(AC = 10\), we have \(BC=\frac{\sqrt{3}}{2}\times10=5\sqrt{3}\) units. We can also use the Pythagorean theorem. In right - triangle \(ABC\), \(AC^{2}=AB^{2}+BC^{2}\). We know \(AC = 10\) and \(AB = 5\), so \(10^{2}=5^{2}+BC^{2}\), \(100 = 25+BC^{2}\), \(BC^{2}=75\), \(BC = 5\sqrt{3}\) (we take the positive value as length cannot be negative).

Answer:

\(AB = \boldsymbol{5}\) units, \(BC=\boldsymbol{5\sqrt{3}}\) units