In triangle ABC right angled at B, AB = 12cm and ∠CAB = 60°. Determine the length of side BC.

 12 × √3 cm

In triangle ABC, which is right-angled at B, we are given the following information:

• AB = 12 cm (length of side AB)
• ∠CAB = 60° (angle at vertex A)
• ∠ABC = 90° (right angle at vertex B)

We are tasked with determining the length of side BC. To solve this, we can use trigonometric ratios, specifically the sine and cosine functions, as the triangle is a right-angled triangle.

Step 1: Understand the relationship between the sides and angles in a right-angled triangle.

In a right-angled triangle, the trigonometric ratios like sine, cosine, and tangent are used to relate the angles to the sides of the triangle. Here, we are given the angle ∠CAB = 60° and the side AB, which is adjacent to this angle.

Step 2: Apply the tangent function.

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In our case, angle ∠CAB = 60° is opposite side BC, and side AB is adjacent to this angle. The tangent of 60° is known to be √3 (approximately 1.732).

Step 3: Set up the equation using the tangent function.

We know that:

tan(60°) = BC / AB

Substitute the known values:

√3 = BC / 12

Step 4: Solve for BC.

Multiply both sides of the equation by 12 to find BC:

BC = 12 × √3

The length of side BC is 12 × √3 cm, which matches option a) 12 × √3 cm.

Therefore, the correct answer is option a) 12 × √3 cm.