The perimeter of a triangle is 60 cm. The longest side exceeds the shortest one by 10 cm.and the sum of the lengths of the longest and the shortest ones is twice the length of the other side. Find the lengths of the sides.

There's a typo in the in the fourth line from the bottom.It should be corrected as,

4x +20 +2x +10 = * 120

* is to indicate the place I've corrected

Let the shortest side be x(cm).So the longest side equals to (x+10)cm and the other side equals to (x+x+10)/2

So perimeter of the triangle= x + (x+10)+[(x+x+10)/2] =60
2x+10+[(2x+10)/2] =60

Now let's multiply both sides by 2 to get,
2(2x+10) + (2x+10) = 60*2
4x+20 +2x+10 = 129
6x+30=120
6x=90
x= 15 cm

Let's denote the lengths of the shortest, middle, and longest sides as x, y, and z, respectively.

According to the given information, the longest side (z) exceeds the shortest side (x) by 10 cm:
z = x + 10

Moreover, the sum of the lengths of the longest and the shortest sides (z + x) is twice the length of the middle side (y):
z + x = 2y

We know that the perimeter of a triangle is equal to the sum of all its sides:
Perimeter = x + y + z = 60 cm

Now we can solve this system of equations to find the lengths of the sides of the triangle.

From the equation z + x = 2y, we can express x in terms of y:
x = 2y - z

Substituting this expression for x into the equation for the perimeter, we get:
(2y - z) + y + z = 60
3y = 60
y = 20 cm

Substituting the value of y back into the equation z = x + 10, we get:
z = x + 10
z = 20 + 10
z = 30 cm

Finally, substituting the values of y = 20 cm and z = 30 cm into the equation x = 2y - z, we can solve for x:
x = 2y - z
x = 2(20) - 30
x = 40 - 30
x = 10 cm

Therefore, the lengths of the sides of the triangle are x = 10 cm, y = 20 cm, and z = 30 cm.

To solve this problem, we can start by assigning variables to the lengths of the three sides of the triangle. Let's call the lengths of the shortest, middle, and longest sides as x, y, and z, respectively.

From the given information, we can form two equations:

1. The longest side exceeds the shortest one by 10 cm:
z = x + 10

2. The sum of the lengths of the longest and shortest ones is twice the length of the other side:
z + x = 2y

We also know that the perimeter of a triangle is the sum of the lengths of all its sides. So, we can write the equation for the perimeter as follows:

Perimeter = x + y + z

From the given information, we know that the perimeter of the triangle is 60 cm. Hence:

60 = x + y + z

Now, we have a system of three equations with three variables (x, y, and z). We can solve this system using substitution or elimination method.

Let's solve this system using substitution method:

Substituting the value of z from equation (1) into equation (2), we get:

(x + 10) + x = 2y
2x + 10 = 2y
y = x + 5

Substituting the values of y and z into the equation for the perimeter, we get:

60 = x + (x + 5) + (x + 10)
60 = 3x + 15
3x = 45
x = 15

Now, we can find the values of y and z:

y = x + 5 = 15 +5 = 20

z = x + 10 = 15 + 10 = 25

Therefore, the lengths of the sides of the triangle are 15 cm, 20 cm, and 25 cm.