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 17 inches and a second side that is 1 inch less than twice the thrid side, what are the possible lengths for the second and third side.
it seems to me that the third side has to be greater then sixteen, and less than 18.
Two sides of a triangle are 11cm and 16cm what is the range of possible lengths for the third side
Let's assume the lengths of the second and third sides are represented by variables x and y, respectively.
Given that the sum of the lengths of any two sides of a triangle must be greater than the third side, we can set up the following inequality:
x + y > 17 (Inequality 1)
We also know that the second side is 1 inch less than twice the third side, so we can write:
x = 2y - 1 (Equation 2)
To find the possible lengths for the second and third sides, we need to solve this system of equations.
Substituting the value of x from Equation 2 into Inequality 1, we have:
(2y - 1) + y > 17
3y - 1 > 17
3y > 18
y > 6
So, y must be greater than 6.
Now, let's find the corresponding value of x using Equation 2:
x = 2y - 1
x = 2(6) - 1
x = 11
Therefore, if the third side (y) is greater than 6, the second side (x) must be 11.
To solve this problem, we will use the given information regarding the triangle's sides. Let's represent the lengths of the three sides as follows:
Side 1: 17 inches
Side 2: x inches (second side)
Side 3: y inches (third side)
From the information given, we know that Side 2 is 1 inch less than twice Side 3. So we can write this as an equation:
x = 2y - 1
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.
So, using this inequality, we can create three equations to consider:
1. Side 1 + Side 2 > Side 3
2. Side 1 + Side 3 > Side 2
3. Side 2 + Side 3 > Side 1
Substituting the given values, we have:
1. 17 + x > y
2. 17 + y > x
3. x + y > 17
Now, let's substitute the value of x from the given equation:
x = 2y - 1
We have:
1. 17 + (2y - 1) > y
2. 17 + y > 2y - 1
3. (2y - 1) + y > 17
Let's solve each equation step-by-step:
1. 17 + 2y - 1 > y
16 + 2y > y
16 > -y
-16 < y
2. 17 + y > 2y - 1
18 > y
3. 3y - 1 > 17
3y > 18
y > 6
From the calculations, we find:
-16 < y
y > 6
Therefore, the possible lengths for the second side (Side 2) and the third side (Side 3) are:
-16 < y
y > 6
Note: The negative value for y (-16 < y) indicates that the length cannot be negative, so we exclude the negative range. Therefore, the possible lengths for Side 2 and Side 3 are y > 6.
In conclusion, the second side (Side 2) must be greater than 6 inches, and the third side (Side 3) must be greater than 6 inches as well.