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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.

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)

## If two triangles have the three sides of the one equal to the three sides of the other, each to each, then they are equal in all respects.

If two triangles have the three sides of the one equal to the three sides ...