# Find the length of a circle inscribed in an isosceles right-angled triangle with a hypotenuse c.

January 28, 2021 | education

| Since the triangle ABC is rectangular and isosceles, then 2 * AC ^ 2 = 2 * BC ^ 2 = AB ^ 2.

AC = BC = AB / √2 = C / √2 cm.

The radius of the inscribed circle in a right-angled triangle is equal to half the sum of its leg lengths minus the length of the hypotenuse.

R = (AC + BC – AB) / 2 = (С / √2 + С / √2 – С) / 2 = (2 * С / √2 – С) / 2 = (С * √2 – С) / 2 = C * (√2 – 1) / 2 cm.

Then the length of the inscribed circle is: L = 2 * π * R = 2 * π * С * (√2 – 1) / 2 = π * С * (√2 – 1) cm.

Answer: The circumference is π * С * (√2 – 1) cm.

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