In a ∆ABC, angle BAC = 30° and angle BCA = 60°. If AC = 13 cm and AB = 12 cm, then BC is equal to:

The correct answer is 5 cm.

To find the length of the side BC in triangle ABC, let us analyze the given information step-by-step:

Step 1: Determine the measure of the third angle of the triangle.
We are given two angles of the triangle ABC:
Angle BAC = 30°
Angle BCA = 60°
Since the sum of the angles in any triangle is always 180°, we can calculate the measure of angle ABC as follows:

$\angle ABC=180°-(\angle BAC+\angle BCA)$

$\angle ABC=180°-(30°+60°)$

$\angle ABC=180°-90°=90°$

Since angle ABC is 90°, triangle ABC is a right-angled triangle with the right angle at vertex B.

Step 2: Identify the sides of the right-angled triangle.
In a right-angled triangle, the side opposite to the 90° angle is the hypotenuse. Here, the side opposite to angle ABC is AC.
So, the hypotenuse is AC = 13 cm.
The other two sides are AB = 12 cm and the unknown side BC.

Step 3: Apply Pythagoras' Theorem.
According to Pythagoras' theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

$A{C}^2=A{B}^2+B{C}^2$

Substituting the given values into the equation:

${13}^2={12}^2+B{C}^2$

$169=144+B{C}^2$

Subtract 144 from both sides to isolate the variable:

$B{C}^2=169-144$

$B{C}^2=25$

Taking the square root of both sides:

$BC=\sqrt{25}=5\text{cm}$

Therefore, the length of BC is 5 cm.