In right angled triangle ABC, ∠B = 90° and AC is the hypotenuse. If m∠A = 30° and AC = 10 cm, then what is the length of side AB (in cm)?

Introduction / Context:
This question uses basic trigonometry in a right angled triangle, specifically the properties of a 30°–60°–90° triangle. In such special triangles, the ratios of the sides are fixed: the side opposite 30° is half the hypotenuse, and the side opposite 60° is (√3 / 2) times the hypotenuse. Alternatively, you can use trigonometric functions such as sine and cosine to relate side lengths and angles. Problems like this are common in aptitude tests because they check whether you can quickly apply these well known ratios or simple trigonometric formulas.

Given Data / Assumptions:
• Triangle ABC is right angled at B, so ∠B = 90°.
• AC is the hypotenuse of the triangle and AC = 10 cm.
• Angle at A is ∠A = 30°.
• Hence angle at C is ∠C = 60° by the angle sum property of a triangle.
• We are required to find the length of side AB in centimetres.

Concept / Approach:
In a right triangle, the side opposite the right angle is the hypotenuse. Here, AC is the hypotenuse. Side AB is adjacent to angle A and side BC is opposite to angle A. Using cosine, cos(A) = adjacent / hypotenuse = AB / AC. For angle A = 30°, cos(30°) = √3 / 2. So AB = AC * cos(30°) = 10 * (√3 / 2). This gives the required length directly. Another way is to use the standard 30°–60°–90° ratio, where sides relative to the hypotenuse follow fixed proportions.

Step-by-Step Solution:
Step 1: In right triangle ABC, identify AC as the hypotenuse and AB as the side adjacent to angle A. Step 2: Write the cosine relation: cos(A) = AB / AC. Step 3: Substitute A = 30° and AC = 10 cm into the relation: cos(30°) = AB / 10. Step 4: Recall that cos(30°) = √3 / 2. Step 5: Substitute this value: √3 / 2 = AB / 10. Step 6: Solve for AB: AB = 10 * (√3 / 2) = 5√3 cm.

Verification / Alternative check:
We can also use the standard side ratio for a 30°–60°–90° triangle. The side opposite 30° (here BC) is half the hypotenuse, so BC = 10 / 2 = 5 cm. The side opposite 60° (here AB) is BC * √3 = 5 * √3, which agrees with the previous result. Numerically, 5√3 is about 5 * 1.732 ≈ 8.66 cm, which is less than the hypotenuse 10 cm and greater than the shorter leg 5 cm, so the lengths are in the correct order.

Why Other Options Are Wrong:
Option 5 cm is actually the length of the side opposite 30°, not the side adjacent, so it represents BC, not AB. Option 10√3 cm is larger than the hypotenuse and therefore cannot be a leg of the triangle. Option 10 cm is the hypotenuse itself, not one of the legs. Thus only 5√3 cm correctly matches the geometric and trigonometric relationships in the given triangle.

Common Pitfalls:
Students sometimes mix up which side is opposite and which is adjacent to the given angle, leading them to use sine instead of cosine or vice versa inappropriately. Another frequent error is to confuse the special triangle ratios and assume the side opposite 30° is (√3 / 2) times the hypotenuse, which is incorrect. Carefully drawing a quick diagram, labelling the angles and sides, and recalling that the side opposite 30° is the shortest leg can help ensure that you choose the correct formula and side ordering.

Final Answer:
The length of side AB is 5√3 cm.