The perimeter of an isosceles triangle is 20 cm and if b is the length of the third side which is distinct from the other two equal sides, find b.

Options

A. `((1,20/3))cup((20/3,10))`
B. `((0,20/3))cup((20/3,10))`
C. `0 lt b lt 10`
D. `5 lt b lt 10`

Answer

To find the length of the third side \( b \) of an isosceles triangle with a perimeter of 20 cm, we can follow these steps:

### Step 1: Understand the properties of the isosceles triangle
In an isosceles triangle, two sides are equal. Let's denote the length of the two equal sides as \( a \) and the length of the distinct side as \( b \).

### Step 2: Write the equation for the perimeter
The perimeter \( P \) of the triangle is given by the sum of all its sides:
\[ P = a + a + b = 2a + b \]
Given that the perimeter is 20 cm, we can set up the equation:
\[ 2a + b = 20 \]

### Step 3: Rearrange the equation to find \( b \)
We can rearrange the equation to express \( b \) in terms of \( a \):
\[ b = 20 - 2a \]

### Step 4: Apply the triangle inequality
For a triangle to exist, it must satisfy the triangle inequality theorem. This means:
1. The sum of the lengths of any two sides must be greater than the length of the third side.

For our isosceles triangle, we need to consider the following inequalities:
1. \( a + a > b \) (or \( 2a > b \))
2. \( a + b > a \) (which simplifies to \( b > 0 \))
3. \( a + b > a \) (which is always true since \( b > 0 \))

### Step 5: Substitute \( b \) into the inequalities
From the first inequality \( 2a > b \):
\[ 2a > 20 - 2a \]
Adding \( 2a \) to both sides gives:
\[ 4a > 20 \]
Dividing by 4:
\[ a > 5 \]

From the second inequality \( b > 0 \):
\[ 20 - 2a > 0 \]
Adding \( 2a \) to both sides gives:
\[ 20 > 2a \]
Dividing by 2:
\[ 10 > a \]

### Step 6: Combine the inequalities
We now have two inequalities for \( a \):
\[ 5 < a < 10 \]

### Step 7: Determine the range for \( b \)
Now, we can find the range for \( b \) using the expression \( b = 20 - 2a \):
- When \( a \) is slightly greater than 5 (let's say \( a = 5.1 \)):
\[ b = 20 - 2(5.1) = 20 - 10.2 = 9.8 \]
- When \( a \) is slightly less than 10 (let's say \( a = 9.9 \)):
\[ b = 20 - 2(9.9) = 20 - 19.8 = 0.2 \]

Thus, \( b \) must be in the range:
\[ 0 < b < 10 \]

### Final Answer
The length of the third side \( b \) can be any value in the range \( (0, 10) \).