In triangle ABC, angle B is a right angle (∠B = 90°). Point D is the midpoint of hypotenuse AC. If AB = 6 cm and BC = 8 cm, what is the length (in cm) of segment BD?

Introduction / Context:
This problem uses a beautiful property of right angled triangles involving the midpoint of the hypotenuse. It tests your understanding that in a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices. This geometric fact allows you to find distances quickly without heavy calculations.

Given Data / Assumptions:

• Triangle ABC is right angled at B (∠B = 90°).
• AB = 6 cm, BC = 8 cm.
• AC is the hypotenuse.
• D is the midpoint of AC.
• We must find the length BD.

Concept / Approach:
In any right angled triangle, the midpoint of the hypotenuse is equidistant from all three vertices. That is: DA = DB = DC = (hypotenuse) / 2 So, if we first compute the hypotenuse AC using the Pythagoras theorem, then BD is simply half of AC. This property can be remembered as: the midpoint of the hypotenuse is the circumcentre of the right triangle, and the circumradius equals half the hypotenuse.

Step-by-Step Solution:
Step 1: Use the Pythagoras theorem in triangle ABC to find AC. Step 2: AC^2 = AB^2 + BC^2 = 6^2 + 8^2 = 36 + 64 = 100. Step 3: Therefore, AC = √100 = 10 cm. Step 4: D is the midpoint of AC, so AD = DC = AC / 2 = 10 / 2 = 5 cm. Step 5: By the right triangle midpoint property, BD = AD = DC = 5 cm. Step 6: Hence, the required length BD is 5 cm.

Verification / Alternative check:
We can verify using coordinates. Place B at (0, 0), A at (6, 0) and C at (0, 8). Then AC is the segment from (6, 0) to (0, 8). Its midpoint D is ((6 + 0) / 2, (0 + 8) / 2) = (3, 4). The distance from B to D is √(3^2 + 4^2) = √(9 + 16) = √25 = 5 cm, which matches the result BD = 5 cm and confirms the geometric property.

Why Other Options Are Wrong:
4 cm and 8 cm do not equal half the hypotenuse and do not match the circumradius of the right triangle. 12 cm is larger than the hypotenuse itself, which is 10 cm, so it cannot be a distance from a vertex to a point on the hypotenuse. 10 cm is the length of the hypotenuse AC, not the distance from B to the midpoint of AC. Thus none of these values are consistent with the geometry of a right triangle and the midpoint property.

Common Pitfalls:
A frequent mistake is to treat BD as the median and try to apply the median length formula inconsistently, instead of recalling the special right triangle property. Another error is computing AC correctly but then forgetting that BD equals half of AC in this specific case. Remembering that the midpoint of the hypotenuse is the centre of the circumscribed circle around the right triangle helps to quickly recall that all three vertices are the same distance from this point.

Final Answer:
The length of BD is 5 cm.