bsaikrishna wrote:

If a right angled triangle is inscribed in a circle, is it necessary for the hypotenuse of the right triangle to be the diameter of the circle? I know that the vice-versa is true.

You can figure this out by drawing a diagram.

Attachment:

Ques3.jpg [ 9.44 KiB | Viewed 10413 times ]
Look how the angle is increasing as you go higher up. Hence for every length of the minor arc, there is a unique inscribed and central angle. The right triangle's hypotenuse will be the largest length of the chord i.e. a diameter and its central angle will be 180 giving the inscribed angle as 90.

Also, length of arc = (Central angle/360) * 2*pi*r

Since inscribed angle is 90, central angle is 180.

length of arc = 180/360 * (2*pi*r)

length of arc = pi*r

i.e. you get a semi circle. So the chord (the hypotenuse of the right triangle) must be the diameter.

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Karishma

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