The perimeter of a triangle is 19 cm. If the length of the longest side is twice that of the shortest side and 3 cm less than the sum of the lengths of the other two sides, find the lengths of the three sides

If the shortest side is x and the middle length is y, then

x + y + 2x = 19

But 2x is also x+y-3, so y=x+3, and

x + x+3 + 2x = 19
x = 4

and the sides are 4,7,8

To solve this problem, let's assign variables and use the given information to form equations. Let's call the lengths of the triangle's sides: a, b, and c.

According to the problem, the perimeter is 19 cm. The perimeter of a triangle is the sum of the lengths of its three sides, so we can write the equation:

a + b + c = 19 (equation 1)

The problem also states that the length of the longest side is twice that of the shortest side. We can express this as an equation:

c = 2a (equation 2)

Additionally, it is mentioned that the longest side is 3 cm less than the sum of the lengths of the other two sides. This can be written as:

c = a + b - 3 (equation 3)

Now we have a system of equations (equations 1, 2, and 3) that we can solve simultaneously.

To eliminate the variable c, we can substitute equation 2 into equation 3:

2a = a + b - 3

Rearranging the equation:

a - b = -3 (equation 4)

Now we have a system of two equations (equations 1 and 4) with two variables (a and b) that we can solve simultaneously.

From equation 1, we can express b in terms of a:

b = 19 - a - c

Substituting equation 2:

b = 19 - a - 2a

Simplifying:

b = 19 - 3a (equation 5)

Next, we can substitute equation 5 into equation 4:

a - (19 - 3a) = -3

Expanding the equation:

a - 19 + 3a = -3

Combining like terms:

4a - 19 = -3

Adding 19 to both sides:

4a = 16

Dividing both sides by 4:

a = 4

Using equation 5, we can find the value of b:

b = 19 - 3a

b = 19 - 3(4)

b = 19 - 12

b = 7

Finally, we can find the value of c using equation 2:

c = 2a

c = 2(4)

c = 8

Therefore, the lengths of the three sides of the triangle are 4 cm, 7 cm, and 8 cm.

Let's use variables to represent the lengths of the sides of the triangle.

Let's say the lengths of the three sides are a, b and c (with a being the shortest side).

According to the given information:
1) The perimeter of the triangle is 19 cm.
- We can write this as: a + b + c = 19.

2) The length of the longest side is twice that of the shortest side.
- This can be written as: c = 2a.

3) The length of the longest side is 3 cm less than the sum of the lengths of the other two sides.
- This can be written as: c = a + b - 3.

Since we have two equations for c, we can set them equal to each other:
2a = a + b - 3.

Simplifying this equation, we get:
a = b + 3.

Now we have a system of two equations:
1) a + b + c = 19,
2) a = b + 3.

We can substitute the value of "a" from equation 2 into equation 1:
(b + 3) + b + c = 19.

Simplifying further:
2b + c = 16.

Since we have three variables and two equations, we need another equation to solve for the lengths of the sides. However, we don't have enough information to determine the exact lengths of the sides.

Additional information will be required to solve the problem and find the lengths of the three sides.