Draw a right triangle and the perpendicular from the midpoint of the hypotenuse to the base. i) Prove that this perpendicular is half the perpendicular side of the large triangle. ii) Prove that in the large triangle, the distance from the midpoint of the hypotenuse to all the vertices are equal. iii) Prove that the circumcentre of a right triangle is the midpoint of its hypotenuse.

Question

Philoid · Accepted 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.