an isosceles triangle R has side length 17 , 17 and 16. Region S consist of the set of points inside of R that are a distance of at least 2 from the sides of R . The area of S divided by the area of R is m/n ,where m and n are relatively prime positive integers . find m+n

To solve the problem, we need to find the ratio of the area of region \( S \) to the area of triangle \( R \), where region \( S \) is the set of points inside \( R \) that are at least 2 units away from all sides.

### Step 1: Calculate the area of triangle \( R \)
Given that the sides of the triangle are 17, 17, and 16, we can use Heron's formula to find the area of \( R \).

First, calculate the semi-perimeter \( s \) of the triangle:
\[
s = \frac{17 + 17 + 16}{2} = 25
\]

Now, apply Heron's formula:
\[
\text{Area of } R = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( a = 17 \), \( b = 17 \), and \( c = 16 \).

\[
\text{Area of } R = \sqrt{25(25-17)(25-17)(25-16)} = \sqrt{25 \times 8 \times 8 \times 9} = \sqrt{14400} = 120
\]

### Step 2: Determine the area of region \( S \)
Region \( S \) is formed by shrinking the original triangle \( R \) by a distance of 2 units from each side. This operation forms a smaller, similar triangle inside \( R \).

The area of the smaller triangle \( S \) is related to the area of triangle \( R \) by the square of the ratio of their corresponding side lengths. Since the new triangle's sides are each reduced by 2 units from the original sides, we need to calculate the reduction ratio.

However, since the reduction applies uniformly around the perimeter, the problem simplifies to finding the ratio based on the inradius.

### Step 3: Calculate the inradius of the original triangle
The inradius \( r \) of the original triangle \( R \) is given by:
\[
r = \frac{\text{Area of } R}{s} = \frac{120}{25} = 4.8
\]

When we shrink the triangle by 2 units, the inradius of the smaller triangle \( S \) becomes:
\[
r' = 4.8 - 2 = 2.8
\]

### Step 4: Calculate the ratio of the areas
The ratio of the areas of the smaller triangle to the original triangle is the square of the ratio of their inradii:
\[
\frac{\text{Area of } S}{\text{Area of } R} = \left(\frac{r'}{r}\right)^2 = \left(\frac{2.8}{4.8}\right)^2 = \left(\frac{7}{12}\right)^2 = \frac{49}{144}
\]

Thus, the area of region \( S \) divided by the area of region \( R \) is \( \frac{49}{144} \).

### Step 5: Find \( m+n \)
Here, \( m = 49 \) and \( n = 144 \), so \( m+n = 49 + 144 = 193 \).

Thus, the final answer is:
\[
\boxed{193}
\]

Would you like a deeper explanation or have any questions? Here are some related problems to consider:

1. How do you find the area of a triangle using the sine rule?
2. What is the significance of the inradius in determining the area of a triangle?
3. How would the result change if the sides of the triangle were different?
4. How can Heron's formula be applied to other polygons?
5. How does shrinking a triangle by a constant distance affect its centroid?

**Tip:** When dealing with similar triangles, remember that the ratio of their areas is the square of the ratio of their corresponding sides.

Learn how to calculate the area ratio of region S inside an isosceles triangle R with sides 17, 17, and 16. Understand Heron's formula, the concept of inradius, and how to find the ratio of areas of similar triangles. Suitable for high school students studying geometry and triangle properties.

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