(a) Let be a triangle with right angle and hypotenuse .
(See the figure.)
If the inscribed circle touches the hypotenuse at D,
show that .
(b) If , express the radius of the inscribed circle in terms of and . (See the figure.)
If the inscribed circle touches the hypotenuse at D,
show that .
(c) If is fixed and varies, find the maximum value of .
Solution
Let be the centre of the circle and and be points of
tangency of and respectively.
(given)
Draw and .
Since and , is a square.
Therefore
Let .
(a)
(b) Draw . Since is incentre, bisects .
In right ,
In right , By Pythagoras Theorem,
(c) Since is fixed and varies, is a function of and the rate of
change of with respect to is
has stationary value when .
and .
When ,
.
will be maximum value when .
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