Draw right-angled \(\Delta ABC\) of lengths of sides are 3 cm, 4 cm and 5 cm. Draw median on the hypotenuse of \(\Delta ABC\). Then :(i) Measure the length of median and write it.(ii) By observing lengths of median and hypotenuse write your observati…

Answer: (i) the median on the hypotenuse is 2.5 cm (ii) the median drawn to the hypotenuse of a right-angled triangle is half of the hypotenuse.

Construction:

1. Draw seg BC = 3 cm.
2. At B draw ray BX \(\perp\) BC (\(\angle B = 90^\circ\)).
3. On ray BX mark A such that BA = 4 cm. Join AC; on measuring, AC = 5 cm (hypotenuse).
4. Draw the perpendicular bisector of AC; it meets AC at its midpoint D (AD = DC = 2.5 cm).
5. Draw seg BD. Seg BD is the median on the hypotenuse.

Final figure (as drawn): right-angled \(\triangle ABC\) with \(\angle B = 90^\circ\) marked, AB = 4 cm, BC = 3 cm, AC = 5 cm; the bisecting arcs on AC; midpoint D; seg BD.

(i) On measuring, BD = 2.5 cm.

(ii) AC = 5 cm and BD = 2.5 cm, so \(BD = \dfrac12 AC\). Observation: in a right-angled triangle, the median drawn on the hypotenuse is half the hypotenuse (BD = AD = DC, so D is equidistant from all three vertices).

(Note: a picture cannot be shown here; in the examination the figure must actually be drawn with ruler and compass, labelled as described, and the construction arcs must be kept.)