Median of Right triangle property

The property states that ” in right triangle, the length of median that touches hypotenuse is half of the length of hypotenuse “.

Length of median of right triangle

Consider the above right triangle ABC in which ∠ABC = 90 degree.

Here BM is the median of triangle which divide the hypotenuse into two equal parts, AM = MC.

According to the theorem, the length of median BM is half of hypotenuse AC.

i.e. \mathtt{BM=\frac{1}{2} .AC}

Proof of median of right triangle property

Length of median of right triangle is half of hypotenuse

Given above is right triangle ABC.
∠ABC = 90 degree

BM is the median intersecting the hypotenuse AC.
i.e. AM = MC.

Draw line MN such that it is parallel to line BC.
i.e. MN || BC

To prove:
Median is half of hypotenuse.
\mathtt{BM=\frac{1}{2} .AC}

Consider triangle ABC.

M is the midpoint of AC { given }
Line MN || BC { construction }

Using the midpoint theorem of triangle, we can say that point N is midpoint of AB.
i.e. AN = NB

We know that NM || BC.

Since, BC is perpendicular to AB. We can say that NM is also perpendicular to AB.

i.e. ∠ANM = 90 degree.

Take triangle ANM and BNM

AN = NB { Proved above }
NM = MN { common side }
∠ANM = ∠BNM = 90 degree

By SAS congruency condition we can say that both triangles are congruent.
Hence, \mathtt{\triangle ANM\cong \triangle BNM}

Since both triangles are congruent, we can say that;

This can also be written as; \mathtt{BM=\frac{1}{2} .AC}

Hence, we proved that in right triangle, the length of median of hypotenuse is half of half of hypotenuse.

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