The perimeter of an isosceles triangle is 40 cm. The base is two-third of the sum of equal sides. Find the length of each equal side.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the problem
We have an isosceles triangle with a perimeter of 40 cm. The base is two-thirds of the sum of the equal sides. We need to find the length of each equal side.

### Step 2: Define the variables
Let the length of each equal side be \( x \) cm. Since it is an isosceles triangle, the two equal sides are \( AB \) and \( BC \), both equal to \( x \) cm. The base \( AC \) can be expressed in terms of \( x \).

### Step 3: Express the base in terms of \( x \)
According to the problem, the base \( AC \) is two-thirds of the sum of the equal sides. The sum of the equal sides is \( x + x = 2x \). Therefore, the base \( AC \) can be calculated as:
\[ AC = \frac{2}{3} \times (2x) = \frac{4x}{3} \text{ cm} \]

### Step 4: Write the equation for the perimeter
The perimeter of the triangle is the sum of all its sides. Thus, we can write:
\[ \text{Perimeter} = AB + BC + AC = x + x + \frac{4x}{3} \]
This simplifies to:
\[ 2x + \frac{4x}{3} \]

### Step 5: Set up the equation
We know the perimeter is 40 cm, so we set up the equation:
\[ 2x + \frac{4x}{3} = 40 \]

### Step 6: Solve the equation
To solve this equation, we first find a common denominator for the terms on the left side. The common denominator for 1 and 3 is 3. We can rewrite \( 2x \) as \( \frac{6x}{3} \):
\[ \frac{6x}{3} + \frac{4x}{3} = 40 \]
Combine the fractions:
\[ \frac{6x + 4x}{3} = 40 \]
This simplifies to:
\[ \frac{10x}{3} = 40 \]

### Step 7: Multiply both sides by 3
To eliminate the fraction, multiply both sides by 3:
\[ 10x = 120 \]

### Step 8: Divide by 10
Now, divide both sides by 10 to find \( x \):
\[ x = \frac{120}{10} = 12 \text{ cm} \]

### Step 9: Conclusion
The length of each equal side is \( 12 \) cm.