Certainly! Let's solve the problem step-by-step based on the given information.
### Step-by-Step Solution:
#### Part a: Calculating the sides and [tex]\( S \)[/tex]
1. Understanding the ratio:
The sides of the triangle are in the ratio 6:9:10. Let's denote the sides as [tex]\( 6x \)[/tex], [tex]\( 9x \)[/tex], and [tex]\( 10x \)[/tex], where [tex]\( x \)[/tex] is a common multiplier.
2. Given perimeter:
The perimeter of the triangle is [tex]\( 2500 \, \text{m} \)[/tex].
Thus, we have:
[tex]\[
6x + 9x + 10x = 2500
\][/tex]
3. Solving for [tex]\( x \)[/tex]:
Combine the terms on the left-hand side:
[tex]\[
25x = 2500
\][/tex]
Divide both sides by 25:
[tex]\[
x = \frac{2500}{25} = 100
\][/tex]
4. Calculating the sides:
Using [tex]\( x = 100 \)[/tex]:
[tex]\[
a = 6x = 6 \times 100 = 600 \, \text{m}
\][/tex]
[tex]\[
b = 9x = 9 \times 100 = 900 \, \text{m}
\][/tex]
[tex]\[
c = 10x = 10 \times 100 = 1000 \, \text{m}
\][/tex]
5. Calculating the semi-perimeter [tex]\( S \)[/tex]:
The semi-perimeter [tex]\( S \)[/tex] is half of the perimeter:
[tex]\[
S = \frac{\text{Perimeter}}{2} = \frac{2500}{2} = 1250 \, \text{m}
\][/tex]
Therefore, the value of [tex]\( S \)[/tex] is:
[tex]\[
S = 1250 \, \text{m}
\][/tex]
#### Part b: Perimeter formula
The formula to calculate the perimeter [tex]\( P \)[/tex] of a triangle is the sum of its three sides. For a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
P = a + b + c
\][/tex]
In this problem, the perimeter is already given as [tex]\( 2500 \, \text{m} \)[/tex].
### Summary
- Value of [tex]\( S \)[/tex]:
[tex]\[
S = 1250 \, \text{m}
\][/tex]
- Perimeter formula:
[tex]\[
P = a + b + c
\][/tex]
These are the necessary values and formulas based on the given problem.
