Question

In a triangle $ABC$, $C=90^\circ$. If $r$ is the inradius and $R$ is the circumradius of the triangle, then \[ 2(r+R)= \]

• A. $a+b$
• B. $b+c$
• C. $a+c$
• D. $a+b+c$

Hint:

For a right triangle, remember $R=\frac{\text{hypotenuse}}{2}$ and $r=\frac{a+b-c}{2}$.

Answer 1

The Correct Option is A

Step 1: Concept
Use standard formulas for the inradius and circumradius of a right triangle.

Step 2: Meaning
Since $C=90^\circ$, side $c$ is the hypotenuse. For a right triangle, \[ R=\frac{c}{2} \] and \[ r=\frac{a+b-c}{2}. \]

Step 3: Analysis
Therefore, \[ r+R = \frac{a+b-c}{2} + \frac{c}{2} = \frac{a+b}{2}. \] Multiplying by $2$, \[ 2(r+R)=a+b. \]

Step 4: Conclusion
Hence \[ 2(r+R)=a+b. \]

Final Answer: (A)