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: Understanding the Concept
The problem asks for the sum of the inradius (r) and the circumradius (R) of a right-angled triangle, given the lengths of the two legs.
Step 2: Key Formula or Approach
For a right-angled triangle with legs $a, c$ and hypotenuse $b$: 1. The circumradius $R$ is half the length of the hypotenuse: $R = \frac{b}{2}$. 2. The inradius $r$ can be calculated using the formula $r = \frac{\text{Area}}{\text{semi-perimeter}} = \frac{\Delta}{s}$. A more direct formula for a right triangle is $r = \frac{a+c-b}{2}$. 3. First, we need to find the hypotenuse $b$ using the Pythagorean theorem: $b^2 = a^2 + c^2$.
Step 3: Detailed Explanation
The triangle is right-angled at B. The sides opposite to vertices A, B, C are $a, b, c$ respectively. So, the legs are $a$ and $c$, and the hypotenuse is $b$. Given: $a=13$, $c=84$. 1. Find the hypotenuse b: Using the Pythagorean theorem: \[ b^2 = a^2 + c^2 = 13^2 + 84^2 \] \[ b^2 = 169 + 7056 = 7225 \] \[ b = \sqrt{7225} = 85 \] (Note: $85^2 = (80+5)^2 = 6400 + 800 + 25 = 7225$). 2. Calculate the circumradius R: For a right-angled triangle, the circumcenter is the midpoint of the hypotenuse. \[ R = \frac{b}{2} = \frac{85}{2} = 42.5 \] 3. Calculate the inradius r: Using the formula for a right-angled triangle: \[ r = \frac{a+c-b}{2} = \frac{13+84-85}{2} = \frac{97-85}{2} = \frac{12}{2} = 6 \] 4. Calculate the required sum r + R: \[ r + R = 6 + 42.5 = 48.5 \] Step 4: Final Answer
The sum of the inradius and the circumradius is $r+R = 48.5$.