In an isoceles triangle, the difference between one of the equal sides and the unequal side (longest of the three) is `3//10` of the sum of the equal sides. If the perimeter of the triangle is 90 cm, then find the length of unequal side in centimetres

Options

A. 40
B. 80
C. 25
D. 50

Explanation

To solve the problem step by step, let's denote the equal sides of the isosceles triangle as \( A \) cm each, and the unequal side (the longest side) as \( B \) cm.

### Step 1: Write down the information given
We know:
1. The perimeter of the triangle is 90 cm.
2. The difference between one of the equal sides and the unequal side is \( \frac{3}{10} \) of the sum of the equal sides.

### Step 2: Set up the equations
From the perimeter, we can write:
\[ 2A + B = 90 \]

From the information about the difference, we have:
\[ A - B = \frac{3}{10}(2A) \]

### Step 3: Simplify the second equation
The second equation can be simplified:
\[ A - B = \frac{3}{10}(2A) \implies A - B = \frac{3}{5}A \]

### Step 4: Rearranging the second equation
Rearranging gives us:
\[ A - \frac{3}{5}A = B \implies \frac{2}{5}A = B \]

### Step 5: Substitute \( B \) in the perimeter equation
Now, substitute \( B \) in the perimeter equation:
\[ 2A + \frac{2}{5}A = 90 \]

### Step 6: Combine like terms
To combine the terms, convert \( 2A \) into a fraction:
\[ \frac{10}{5}A + \frac{2}{5}A = 90 \implies \frac{12}{5}A = 90 \]

### Step 7: Solve for \( A \)
Now, multiply both sides by \( 5 \):
\[ 12A = 450 \]
Now divide by 12:
\[ A = \frac{450}{12} = 37.5 \text{ cm} \]

### Step 8: Find \( B \)
Now, substitute \( A \) back to find \( B \):
\[ B = \frac{2}{5}A = \frac{2}{5} \times 37.5 = 15 \text{ cm} \]

### Step 9: Find the length of the unequal side
Now, substitute \( A \) back into the perimeter equation to find \( B \):
\[ B = 90 - 2A = 90 - 2 \times 37.5 = 90 - 75 = 15 \text{ cm} \]

### Conclusion
The length of the unequal side \( B \) is 15 cm.