Mathematics
If a, b, c are the sides of a right angled triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by r = .
Circles
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Answer
Let the circle touch the sides BC, CA and AB of the right triangle ABC at points D, E and F respectively,
where BC = a, CA = b and AB = c (as shown in the given figure).
As the lengths of tangents drawn from an external point to a circle are equal
AE = AF, BD = BF and CD = CE
OD ⊥ BC and OE ⊥ CA (∵ tangents is ⊥ to radius)
ODCE is a square of side r
DC = CE = r
AF = AE = AC - EC = b - r and,
BF = BD = BC - DC = a - r
Now,
AB = AF + BF
⇒ c = (b - r) + (a - r)
⇒ c = b + a - 2r
⇒ 2r = a + b - c
⇒ r = .
Hence, proved that r =
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