Question

Let ABC be a triangle right angled at B. If a = 13 and c = 84, then r + R =

• A. 42.5
• B. 169
• C. 98
• D. 48.5

Hint:

For any right-angled triangle, there's a very useful relation: $r+R = \frac{a+c-b}{2} + \frac{b}{2} = \frac{a+c}{2}$. Using this shortcut, you can directly calculate the sum as $(13+84)/2 = 97/2 = 48.5$ without finding the hypotenuse first (although here you need $b$ for both $r$ and $R$ individually). Wait, my formula for $r$ needed $b$. The relation is $2(r+R) = a+c$. Let's check: $r = (a+c-b)/2$, $R=b/2$. $r+R = (a+c-b+b)/2 = (a+c)/2$. Yes, the shortcut works.

Answer 1

The Correct Option is D

Step 1: Calculate the Hypotenuse (b): Since the triangle is right-angled at B, \( b \) is the hypotenuse. \[ b^2 = a^2 + c^2 = 13^2 + 84^2 \] \[ b^2 = 169 + 7056 = 7225 \] \[ b = \sqrt{7225} = 85 \]
Step 2: Calculate Inradius (r) and Circumradius (R): For a right-angled triangle: 1. Inradius \( r = \frac{\text{sum of legs} - \text{hypotenuse}}{2} = \frac{a+c-b}{2} \) \[ r = \frac{13+84-85}{2} = \frac{12}{2} = 6 \] 2. Circumradius \( R = \frac{\text{hypotenuse}}{2} = \frac{b}{2} \) \[ R = \frac{85}{2} = 42.5 \]
Step 3: Calculate \( r+R \): \[ r + R = 6 + 42.5 = 48.5 \]