The sum of the lengths of any two sides of a triangle must be greater than the third side. If a triangle has one side that is 22 cm and a second side that is 2 cm less than twice the third side, what are the possible lengths for the second and third sides?
let sides be a,b,c
let then b =2c-2
but b+c>22
2c-2+c>22
3c>24
c>8
if c=10 (greater than 8), then b is
b=2c-2=18
the sides are 22,x,2x-2, so we have
2x-2-x < 22 < 2x-2+x
x < 24 < 3x
So, it looks like 8 < x < 24
The length
5cmand25cm.6.2cmand18cm
To find the possible lengths for the second and third sides of the triangle, we can apply the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's assume the length of the third side is x cm.
The second side is described as being 2 cm less than twice the third side. So, the length of the second side is (2x - 2) cm.
Now, we can use the triangle inequality theorem to set up the equation:
22 + (2x - 2) > x
Simplifying the above equation:
22 + 2x - 2 > x
Combine like terms:
2x + 20 > x
Subtract x from both sides:
2x - x + 20 > 0
Simplify:
x + 20 > 0
Subtract 20 from both sides:
x > -20
Since the length of a side cannot be negative in this context, we can conclude that x must be greater than 0.
Therefore, the possible lengths for the second and third sides of the triangle are:
- The second side can have a length of (2x - 2) cm, where x > 0.
- The third side can have a length of x cm, where x > 0.