General enunciation: We have to prove that the line segment joining the middle point of the hypotenuse of a right angle triangle and the opposite vertex is half the hypotenuse.
Particular enunciation: Suppose in right angle triangle ∆ABC, ∠B=900
and AC is hypotenuse. BO is the line joining the middle point of AC and the opposite vertex B of AC. Let us prove that BO=½AC
Construction: Let us draw circle with center at O and OA or OC as radius.
Proof: Since AC is the diameter of the circle then we get ∠ABC=900[∵ Semi-circular angles]
So the three vertices of ∆ABC are on the circumference.
Since O is the centre then OA=OB=OC [Radii of the same circle]
Now we get,
⇒ 2OA =AC
⇒OA = ½AC
∴ BO = ½AC (Proved)