The perimeter of a triangle is 47 km. the first side is 5 km shorter than twice the second. the third side is 2 km longer than the first. find the length of each side.
a = the first side
b = the second side
c = the third side
P = the perimeter
a = 2 b - 5
c = a + 2 = 2 b - 5 + 2 = 2 b - 3
P = a + b + c = 47
a + b + c = 47
2 b - 5 + b + 2 b - 3 = 47
5 b - 8 = 47 Add 8 to both sides
5 b - 8 + 8 = 47 + 8
5 b = 55 Divide both sides by 5
b = 11 km
a = 2 b - 5 = 2 * 11 - 5 = 22 - 5 = 17 km
c = a + 2 = 19 km
P = a + b + c = 17 + 11 + 19 = 47 km
Let's denote the lengths of the second, first, and third sides of the triangle as x, y, and z respectively.
We know that the perimeter of a triangle is the sum of all its sides, so we can create an equation:
x + y + z = 47 ----(1)
From the given information, we can also form the following equations:
y = 2x - 5 ----(2)
z = y + 2 ----(3)
Now, let's substitute the values of y and z from equations (2) and (3) into equation (1):
x + (2x - 5) + ((2x - 5) + 2) = 47
Simplifying this equation, we get:
5x - 3 = 47
Adding 3 to both sides, we get:
5x = 50
Dividing both sides by 5, we find:
x = 10
Now, we can substitute the value of x into equations (2) and (3) to find the values of y and z:
y = 2(10) - 5 = 15
z = 15 + 2 = 17
Therefore, the lengths of the sides of the triangle are:
First side (x) = 10 km
Second side (y) = 15 km
Third side (z) = 17 km
To find the lengths of the sides of the triangle, we can use the information given.
Let's denote the second side as 'x' km. Therefore, the first side is 5 km shorter than twice the second side, which can be written as (2x - 5) km.
The third side is 2 km longer than the first side, so it can be written as (2x - 5 + 2) km, which simplifies to (2x - 3) km.
According to the given information, the perimeter of the triangle is 47 km. The perimeter of a triangle is the sum of the lengths of its sides. Therefore, we can express this information in an equation:
(2x - 5) + x + (2x - 3) = 47
Simplifying the equation, we combine like terms:
5x - 8 = 47
Now, let's solve for 'x':
5x = 47 + 8
5x = 55
x = 55/5
x = 11
So, the second side of the triangle is 11 km.
Using this value, we can find the lengths of the other sides:
First side = 2x - 5 = 2(11) - 5 = 22 - 5 = 17 km
Third side = 2x - 3 = 2(11) - 3 = 22 - 3 = 19 km
Therefore, the lengths of the sides of the triangle are:
First side = 17 km
Second side = 11 km
Third side = 19 km