The perimeter of an isosceles triangle is 32 cm while its equal side stogether measure 18 cm. Find the third side and each of the equal sides.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of the isosceles triangle
An isosceles triangle has two equal sides. Let's denote the lengths of the equal sides as \( a \) and the length of the base (the third side) as \( b \).

### Step 2: Write the formula for the perimeter
The perimeter \( P \) of the triangle can be expressed as:
\[ P = a + a + b = 2a + b \]
According to the problem, the perimeter is 32 cm:
\[ 2a + b = 32 \quad \text{(1)} \]

### Step 3: Use the information about the equal sides
The problem states that the two equal sides together measure 18 cm. Therefore, we can write:
\[ a + a = 2a = 18 \quad \text{(2)} \]

### Step 4: Solve for \( a \)
From equation (2):
\[ 2a = 18 \]
Dividing both sides by 2 gives:
\[ a = 9 \text{ cm} \]

### Step 5: Substitute \( a \) back into the perimeter equation
Now we substitute \( a = 9 \) into equation (1):
\[ 2(9) + b = 32 \]
This simplifies to:
\[ 18 + b = 32 \]

### Step 6: Solve for \( b \)
Subtracting 18 from both sides gives:
\[ b = 32 - 18 = 14 \text{ cm} \]

### Step 7: Summarize the results
The lengths of the sides of the triangle are:
- Each of the equal sides \( a = 9 \) cm
- The third side \( b = 14 \) cm

### Final Answer:
- Each equal side: \( 9 \) cm
- Third side: \( 14 \) cm

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