Question

If \( \triangle ABC \) is a right-angled isosceles triangle and \( \angle C = 90^\circ \), then \( r = \frac{1}{5} \) is:

• A. \( \sqrt{2} + 1 : \sqrt{2} - 1 \)
• B. \( \sqrt{2} - 1 : \sqrt{2} + 1 \)
• C. \( \sqrt{2} : 1 \)
• D. \( 1 : \sqrt{2} \)

Hint:

For right-angled isosceles triangles, use the formula for inradius \( r = \frac{a + b - c}{2} \) and substitute values to calculate the result.

Answer 1

The Correct Option is B

Given that \( \triangle ABC \) is a right-angled isosceles triangle with \( \angle C = 90^\circ \), we know that the two legs are of equal length. Let the length of each leg be \( a \). Then, the hypotenuse \( BC \) will be: \[ BC = a\sqrt{2} \] The formula for the inradius \( r \) of a right-angled triangle is given by: \[ r = \frac{a + b - c}{2} \] where \( a \) and \( b \) are the lengths of the two legs, and \( c \) is the length of the hypotenuse. For our isosceles triangle, \( a = b \), so the formula becomes: \[ r = \frac{2a - a\sqrt{2}}{2} \] Substituting \( a = 5 \) into this formula: \[ r = \frac{2(5) - (5\sqrt{2})}{2} = \frac{10 - 5\sqrt{2}}{2} \] Simplifying this expression, we get: \[ r = \frac{5}{2} \left( 2 - \sqrt{2} \right) \] Thus, the correct answer is option (2).