If the perimeter of a rectangular plot is 34 metres and its area is 60 square metres, what is the length of each of the shorter side?

Options

A. 10m
B. 15m
C. 17m
D. 5m

Answer

To solve the problem step by step, we will use the formulas for the perimeter and area of a rectangle.

### Step 1: Understand the formulas
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
where \( L \) is the length and \( B \) is the breadth.

The area \( A \) of a rectangle is given by:
\[ A = L \times B \]

### Step 2: Set up the equations
From the problem, we know:
1. The perimeter is 34 meters:
\[ 2(L + B) = 34 \]
Dividing both sides by 2:
\[ L + B = 17 \] (Equation 1)

2. The area is 60 square meters:
\[ L \times B = 60 \] (Equation 2)

### Step 3: Express one variable in terms of the other
From Equation 1, we can express \( L \) in terms of \( B \):
\[ L = 17 - B \]

### Step 4: Substitute into the area equation
Now substitute \( L \) in Equation 2:
\[ (17 - B) \times B = 60 \]
Expanding this gives:
\[ 17B - B^2 = 60 \]

### Step 5: Rearrange to form a quadratic equation
Rearranging the equation:
\[ B^2 - 17B + 60 = 0 \]

### Step 6: Factor the quadratic equation
Now we need to factor the quadratic equation:
We look for two numbers that multiply to 60 and add to 17. The numbers are 12 and 5.
Thus, we can factor the equation as:
\[ (B - 12)(B - 5) = 0 \]

### Step 7: Solve for \( B \)
Setting each factor to zero gives us:
1. \( B - 12 = 0 \) → \( B = 12 \)
2. \( B - 5 = 0 \) → \( B = 5 \)

### Step 8: Find the corresponding lengths
Using \( L = 17 - B \):
1. If \( B = 12 \), then \( L = 17 - 12 = 5 \)
2. If \( B = 5 \), then \( L = 17 - 5 = 12 \)

### Conclusion
The lengths of the sides are 12 meters and 5 meters. The shorter side is:
\[ \text{Shorter side} = 5 \text{ meters} \]