In right angled triangle ABC, ∠BAC = 90° and AD is drawn perpendicular to hypotenuse BC. If BD = 7 cm and CD = 28 cm, what is the length (in cm) of altitude AD?

Introduction / Context:
This problem involves a right angled triangle and the altitude drawn from the right angle to the hypotenuse. There is a very useful geometric relation connecting the altitude to the two segments into which it divides the hypotenuse. The question checks whether you know and can apply this standard property quickly in numerical form.

Given Data / Assumptions:

• Triangle ABC is right angled at A, so ∠BAC = 90°.
• AD is perpendicular to BC, so AD is the altitude from the right angle to the hypotenuse.
• BD = 7 cm and CD = 28 cm, where D lies on BC between B and C.
• We need the length of AD in centimetres.
• Triangle is assumed to be non degenerate and right angle is exactly at A.

Concept / Approach:
In a right angled triangle, if an altitude is drawn from the right angle to the hypotenuse, then the square of the altitude length equals the product of the lengths of the two segments into which the hypotenuse is divided. Symbolically, if BD and CD are the segments and AD is the altitude, then AD^2 = BD * CD. This property allows us to compute AD directly from the given segment lengths by simple multiplication and square root extraction.

Step-by-Step Solution:
Step 1: Use the property AD^2 = BD * CD for a right angled triangle with altitude to the hypotenuse. Step 2: Substitute BD = 7 cm and CD = 28 cm, so AD^2 = 7 * 28. Step 3: Compute 7 * 28 = 196. Step 4: Therefore AD^2 = 196. Step 5: Take the positive square root: AD = √196 = 14 cm.

Verification / Alternative check:
We can also verify using Pythagoras theorem. The hypotenuse BC has length √(BD + CD)^2 = 35 cm. There are relations AB^2 = BD * BC and AC^2 = CD * BC in such a configuration. That gives AB^2 = 7 * 35 = 245 and AC^2 = 28 * 35 = 980, so AB = √245 and AC = √980. Pythagoras theorem for triangle ABC would then give AB^2 + AC^2 = 245 + 980 = 1225 = 35^2, which confirms consistency. The altitude formula AD^2 = BD * CD remains valid and yields AD = 14 cm.

Why Other Options Are Wrong:
Values 3.5, 7 and 10.5 correspond to smaller square values that do not equal 196 when squared. The value 21 would produce AD^2 = 441, which is inconsistent with BD * CD = 196. Therefore these options do not satisfy the altitude property in a right angled triangle. Only 14 cm gives the correct relation between the altitude and the hypotenuse segments.

Common Pitfalls:
A typical mistake is to apply Pythagoras theorem directly on BD and CD with AD, which is incorrect because BD and CD are collinear and do not form a right triangle with AD. Another error is to forget the special relation AD^2 = BD * CD and instead attempt a longer coordinate or trigonometric method. Remembering and using standard right triangle properties saves time and reduces algebraic errors in such problems.

Final Answer:
The length of AD is 14 cm.