Analyzing the Right-Angled Triangle

We are given a right-angled triangle ABC with the following properties:

• ∠B = 90°
• ∠ACB = 30°
• Side AB = 5 cm

Since the angles are 90°, 30°, the third angle ∠BAC must be 60° (180° - 90° - 30°). This is a special 30-60-90 triangle.

We need to find the lengths of the hypotenuse (AC) and the adjacent side (BC) relative to the 30° angle, and then calculate their sum (AC + BC).

Calculating Side Lengths Using Trigonometry

We use trigonometric ratios to find the unknown sides.

Finding the Hypotenuse (AC)

The side AB is opposite to ∠ACB. The sine function relates the opposite side and the hypotenuse:

$\sin(\angle ACB) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC}$

Substitute the known values:

$\sin(30°) = \frac{5 \text{ cm}}{AC}$

We know that $\sin(30°) = \frac{1}{2}$.

$\frac{1}{2} = \frac{5}{AC}$

Solving for AC:

AC $= 5 \times 2 = 10$ cm.

Finding the Adjacent Side (BC)

The side BC is adjacent to ∠ACB. The tangent function relates the opposite side and the adjacent side:

$\tan(\angle ACB) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC}$

Substitute the known values:

$\tan(30°) = \frac{5 \text{ cm}}{BC}$

We know that $\tan(30°) = \frac{1}{\sqrt{3}}$.

$\frac{1}{\sqrt{3}} = \frac{5}{BC}$

Solving for BC:

BC $= 5 \times \sqrt{3} = 5\sqrt{3}$ cm.

Final Calculation: AC + BC

Add the calculated lengths of AC and BC:

AC + BC $= 10 \text{ cm} + 5\sqrt{3}$ cm

Therefore, AC + BC $= 10 + 5\sqrt{3}$ cm.