The perimeter of a triangle ABC right angled at C is 70 and the inradius is 6, then `|a-b|=`

Options

A. 1
B. 2
C. 8
D. 9

Explanation

To solve the problem step by step, we will use the properties of a right-angled triangle, the given perimeter, and the inradius.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- The perimeter of triangle ABC is 70.
- The inradius (r) is 6.
- The triangle is right-angled at C.

2. **Calculating the Semi-Perimeter (s):**
\[ s = \frac{\text{Perimeter}}{2} = \frac{70}{2} = 35 \]

3. **Using the Area Formula:**
The area (A) of the triangle can be calculated using the formula:
\[ A = s \times r = 35 \times 6 = 210 \]

4. **Using the Area in Terms of Sides:**
For a right-angled triangle, the area can also be expressed as:
\[ A = \frac{1}{2} \times a \times b \]
Setting the two area expressions equal gives:
\[ \frac{1}{2} \times a \times b = 210 \]
Thus,
\[ a \times b = 420 \]

5. **Using the Perimeter to Relate the Sides:**
From the perimeter, we have:
\[ a + b + c = 70 \]
Therefore,
\[ c = 70 - (a + b) \]

6. **Applying the Pythagorean Theorem:**
Since the triangle is right-angled at C:
\[ a^2 + b^2 = c^2 \]

7. **Expressing \(c^2\) in Terms of \(a\) and \(b\):**
Substitute \(c\) into the Pythagorean theorem:
\[ c = 70 - (a + b) \implies c^2 = (70 - (a + b))^2 \]
Expanding this gives:
\[ c^2 = 4900 - 140(a + b) + (a + b)^2 \]

8. **Relating \(a + b\) and \(ab\):**
We know:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
Therefore, substituting \(c^2\):
\[ a^2 + b^2 = 4900 - 140(a + b) + (a + b)^2 \]
Using \(a^2 + b^2 = c^2\) and \(ab = 420\), we can rearrange and simplify.

9. **Finding \(a - b\):**
We know:
\[ (a - b)^2 = (a + b)^2 - 4ab \]
Substitute \(ab = 420\):
\[ (a - b)^2 = (a + b)^2 - 1680 \]

10. **Final Calculation:**
We can find \(a + b\) from the perimeter:
\[ a + b = 70 - c \]
Substitute \(c\) back into the equations and solve for \(a\) and \(b\).

11. **Finding \(|a - b|\):**
After calculations, we find:
\[ |a - b| = 1 \]

### Final Answer:
\[ |a - b| = 1 \]