Answer :

i) We know that,

In any triangle, line drawn parallel to one side, passing through mid-point of another side will also meet the third side at its mid-point.

Therefore,

Let AD be ‘x’, then DC = x. similarly AE = y then EB = y

By pythagorus theorem for bigger triangle ABC,

----1

Similarly for smaller triangle AED,

-----2

From 1 & 2, we have

Hence proved.

ii)

We have the above following data.

And from pythagorous theorem,

We have

Hence the distance between mid-point of hypotenuse to vertex B (or) A (or) C is x

Hence proved.

iii) We know that

circumcentre is a point from which all the vertices are at equal distances.

In the above case, we have proved the mid-point of hypotenuse is equidistant from all the three vertices. Therefore mid-point of hypotenuse is the circumcentre of the given triangle.

Hence proved.

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