One side of a triangle is half the longest side. The third side is 9 meters less than the longest side. The perimeter is 61 meters. Find all three sides.

c = the longest side

a = c / 2

b = c - 9

P = a + b + c = 61 m

a + b + c = 61

c / 2 + c - 9 + c = 61 Multiply both sides by 2

c + 2 c - 18 + 2 c = 122

5 c - 18 = 122

5 c = 122 + 18

5 c = 140 Divide both sides by 5

c = 140 / 5

c = 28 m

a = c / 2

a = 28 / 2

a = 14 m

b = c - 9

b = 28 - 9

b = 19 m

To solve this problem, we need to set up and solve a system of equations based on the given information.

Let's denote the longest side of the triangle as "x". According to the problem:

"One side of a triangle is half the longest side." This means that one side of the triangle is equal to x/2.
"The third side is 9 meters less than the longest side." This means that the third side is equal to x - 9.

To find the perimeter of the triangle, we add up all three sides:

Perimeter = x + (x/2) + (x - 9) = 61

Now, we can solve this equation for x:

x + (x/2) + (x - 9) = 61
Multiply each term in the equation by 2 to eliminate the fraction:
2x + x + 2(x - 9) = 122
2x + x + 2x - 18 = 122
5x - 18 = 122
Add 18 to both sides of the equation:
5x = 140
Divide both sides of the equation by 5:
x = 28

Now that we have found the value of x, we can substitute it back into the original equations to find the lengths of the other sides.

The first side is half the longest side, so it is (28/2) = 14 meters.
The third side is 9 meters less than the longest side, so it is 28 - 9 = 19 meters.

Therefore, the three sides of the triangle are 14 meters, 28 meters, and 19 meters.

Let's assume that the longest side of the triangle is represented by the variable "x".

According to the problem, one side of the triangle is half the longest side, so this side can be represented as "x/2".

The third side is 9 meters less than the longest side, so it can be represented as "x - 9".

The perimeter of a triangle is the sum of all three sides, which in this case is 61 meters. Therefore, we have the equation:

x + x/2 + (x - 9) = 61

To solve this equation, we'll start by removing the fraction by multiplying the whole equation by 2:

2x + x + 2(x - 9) = 122

Simplifying the equation:

2x + x + 2x - 18 = 122
5x - 18 = 122

Next, we'll isolate the variable by adding 18 to both sides:

5x - 18 + 18 = 122 + 18
5x = 140

Finally, divide both sides of the equation by 5 to solve for "x":

5x/5 = 140/5
x = 28

Now that we know the value of "x", we can find the lengths of all three sides:

Longest side: x = 28 meters
One side: x/2 = 28/2 = 14 meters
Third side: x - 9 = 28 - 9 = 19 meters

Therefore, the three sides of the triangle are 14 meters, 19 meters, and 28 meters.