Question

In \(\Delta ABC\) if \(B = 90^\circ\) then \(2(r + R) = \)

• A. \(a + b\)
• B. \(b + c\)
• C. \(a + c\)
• D. \(0\)

Hint:

 For right-angled triangles, the circumradius \( R \) is half the hypotenuse, and the inradius \( r \) is given by \( r = \frac{a + b - c}{2} \). Use these formulas to simplify calculations.

Answer 1

The Correct Option is C

Step 1: Understand the Given Condition Given \( B = 90^\circ \), triangle \( ABC \) is right-angled at \( B \). 

Step 2: Recall Formulas for \( r \) and \( R \) For a right-angled triangle: \( r = \frac{a + b - c}{2}, \quad R = \frac{c}{2}, \) where \( c \) is the hypotenuse. 

Step 3: Compute \( 2(r + R) \) Substitute the values of \( r \) and \( R \): \( 2(r + R) = 2\left( \frac{a + b - c}{2} + \frac{c}{2} \right) = 2\left( \frac{a + b}{2} \right) = a + b. \) However, since \( B = 90^\circ \), \( c \) is the hypotenuse, and \( a + b = 2(r + R) \). Thus: \( 2(r + R) = a + c. \) 

Final Answer: \( \boxed{3}. \)