In right angled triangle ABC, ∠B = 90° and AC is the hypotenuse. D is the circumcentre of triangle ABC. If AB = 3 cm and BC = 4 cm, what is the length of BD (in cm)?

Introduction / Context:
This question brings together the properties of a right angled triangle and its circumcentre. In any triangle, the circumcentre is the point where the perpendicular bisectors of the sides intersect, and it serves as the centre of the circle passing through all three vertices (the circumcircle). In a right angled triangle, there is a special and very useful fact: the circumcentre lies at the midpoint of the hypotenuse. This allows us to find distances from the circumcentre to the vertices easily using simple length relationships.

Given Data / Assumptions:
• Triangle ABC is right angled at B, so ∠B = 90°.
• AC is the hypotenuse of triangle ABC.
• D is the circumcentre of triangle ABC, that is, the centre of the circumcircle passing through A, B and C.
• AB = 3 cm and BC = 4 cm.
• We need to find the distance BD in centimetres.

Concept / Approach:
In a right angled triangle, the circumcentre lies at the midpoint of the hypotenuse. Therefore, D is the midpoint of AC. The radius of the circumcircle is thus half the hypotenuse length. Furthermore, the circumcentre is equidistant from all three vertices, so BD equals this radius and equals AD and CD. The first step is to compute the hypotenuse AC using the Pythagoras theorem, and then halve it to get the radius and therefore BD.

Step-by-Step Solution:
Step 1: Use the Pythagoras theorem in right triangle ABC to find AC: AC^2 = AB^2 + BC^2. Step 2: Substitute AB = 3 cm and BC = 4 cm: AC^2 = 3^2 + 4^2 = 9 + 16 = 25. Step 3: Take the square root to find AC: AC = √25 = 5 cm. Step 4: In a right angled triangle, the circumcentre lies at the midpoint of the hypotenuse, so D is the midpoint of AC. Step 5: Therefore, the radius of the circumcircle is half the hypotenuse: radius = AC / 2 = 5 / 2 = 2.5 cm. Step 6: Since the circumcentre is equidistant from all three vertices, BD equals the radius, so BD = 2.5 cm.

Verification / Alternative check:
We can verify by noting that AD = CD = BD = 2.5 cm, and D lies exactly midway between A and C. If we place triangle ABC on a coordinate plane with B at the origin, A on the x axis at (3, 0) and C on the y axis at (0, 4), then AC connects (3, 0) to (0, 4). The midpoint of AC is at ((3 + 0) / 2, (0 + 4) / 2) = (1.5, 2). The distance from B (0, 0) to D (1.5, 2) is √(1.5^2 + 2^2) = √(2.25 + 4) = √6.25 = 2.5 cm, confirming our result.

Why Other Options Are Wrong:
If BD were 3 cm or 4 cm, it would equal one of the legs and not half the hypotenuse, which contradicts the known circumcentre property for right triangles. A value of 5.5 cm would even exceed the hypotenuse length of 5 cm, which is impossible for a radius of the circumcircle that must be less than or equal to half the distance between the furthest pair of points. Thus these options are inconsistent with the geometry and the precise calculation of the hypotenuse.

Common Pitfalls:
Students sometimes confuse the circumcentre with the orthocentre, especially in right triangles, and may place the special point at the right angle vertex instead of at the midpoint of the hypotenuse. Another frequent error is to calculate the hypotenuse correctly but forget to divide by two to find the radius and distances from the circumcentre to the vertices. Keeping a clear picture that in a right triangle the circumcentre always lies at the midpoint of the hypotenuse avoids these mistakes and lets you answer such questions very quickly.

Final Answer:
The length of BD is 2.5 cms.