the sum of the lenths of any two sides of a triangle must be greater than the third side. if a triangle has one side that is 17 cm and a second side that is 4cm less than twice the third side, what are the possible lenths for the second and third side?

You succinctly state the property being used. Then you give the two sides. What keeps you from figuring the answer?

Using a,b,c for the sides,
a = 17
b = 2c-4

We know that a+b > c, so 17+2c-4 > c
13 > c

So, any value of c < 13 will work.
For example:
a = 17
c = 10
b = 16

To find the possible lengths of the second and third sides of a triangle, we can use the given information about the lengths of sides.

Let's denote the lengths of the second and third sides as "x" and "y," respectively.

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Mathematically, this can be written as:

x + y > 17 (1)

Additionally, it is given that the second side is 4 cm less than twice the third side. So we can write:

x = 2y - 4 (2)

Now we have two equations (Equations 1 and 2) that we can solve simultaneously to find the possible lengths of the second and third sides.

Substituting Equation 2 into Equation 1, we get:

(2y - 4) + y > 17

Simplifying the equation:

3y - 4 > 17

Adding 4 to both sides:

3y > 21

Dividing both sides by 3:

y > 7

So, the possible lengths for the third side (y) would be any value greater than 7.

Using Equation 2, we can substitute this value of y back into the equation to find the corresponding value for the second side (x):

x = 2(7) - 4
x = 14 - 4
x = 10

Therefore, the possible lengths for the second side (x) would be 10 cm if the third side (y) is greater than 7 cm.

To summarize:

- Third side (y): Any value greater than 7 cm.
- Second side (x): 10 cm.