The permeter of an isosceles triangle is 24 cm. The length of its congruent sides is 13 cm less then twice the length of its base. Find the lengths of all sides of the triangle.

Answer

To solve the problem of finding the lengths of all sides of the isosceles triangle, we can follow these steps:

### Step 1: Define the variables
Let the length of the base of the isosceles triangle be \( x \) cm. Since the triangle is isosceles, the lengths of the two congruent sides will be equal.

### Step 2: Express the lengths of the congruent sides
According to the problem, the length of each of the congruent sides is 13 cm less than twice the length of the base. Therefore, we can express the length of the congruent sides as:
\[ \text{Length of each congruent side} = 2x - 13 \text{ cm} \]

### Step 3: Write the equation for the perimeter
The perimeter of the triangle is the sum of the lengths of all three sides. Given that the perimeter is 24 cm, we can write the equation:
\[ x + (2x - 13) + (2x - 13) = 24 \]

### Step 4: Simplify the equation
Combine like terms in the equation:
\[ x + 2x - 13 + 2x - 13 = 24 \]
\[ 5x - 26 = 24 \]

### Step 5: Solve for \( x \)
Now, we will isolate \( x \) by adding 26 to both sides:
\[ 5x = 24 + 26 \]
\[ 5x = 50 \]
Now, divide both sides by 5:
\[ x = 10 \text{ cm} \]

### Step 6: Find the lengths of the congruent sides
Now that we have the base length \( x \), we can find the lengths of the congruent sides:
\[ \text{Length of each congruent side} = 2(10) - 13 = 20 - 13 = 7 \text{ cm} \]

### Step 7: State the lengths of all sides
Now we can summarize the lengths of all sides of the triangle:
- Base = \( x = 10 \) cm
- Each congruent side = \( 7 \) cm

Thus, the lengths of all sides of the triangle are:
- Base: 10 cm
- Congruent sides: 7 cm, 7 cm

### Final Answer:
The lengths of the sides of the triangle are:
- Base: 10 cm
- Congruent sides: 7 cm each

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