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Home / Geometry / 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.

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.

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.

circle-radii

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,

OA+OC=AC

⇒ OA+OA=AC

⇒ 2OA =AC

⇒OA = ½AC

∴ BO = ½AC              (Proved)

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