Let's solve the problem step by step.

Step 1: Understand the given information

• Area of the rectangle is 540 cm².
• If the length is reduced by 7 cm, the shape becomes a square.

Let's assume:

• The length of the rectangle is $L$ cm.
• The breadth of the rectangle is $B$ cm.

We have two conditions:

1. The area of the rectangle:
   $L \times B = 540 \quad \text{(Equation 1)}$
2. When the length is reduced by 7 cm, the new length becomes equal to the breadth, as it forms a square.
   So:
   $L - 7 = B \quad \text{(Equation 2)}$

Step 2: Substitute Equation 2 into Equation 1

From Equation 2, we know $L = B + 7$. Substitute this into Equation 1:

$(L) \times B = 540$ $(B + 7) \times B = 540$ Expand the equation: $B^2 + 7B = 540$ $B^2 + 7B - 540 = 0$

Step 3: Solve the quadratic equation

We now have a quadratic equation: $B^2 + 7B - 540 = 0$ We will solve it using the quadratic formula. The quadratic formula is given by:

$B = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $B^2 + 7B - 540 = 0$, $a = 1$, $b = 7$, and $c = -540$.

Now, calculate the discriminant:

$\Delta = b^2 - 4ac = 7^2 - 4(1)(-540) = 49 + 2160 = 2209$

Now, apply the quadratic formula: $B = \frac{-7 \pm \sqrt{2209}}{2} = \frac{-7 \pm 47}{2}$

We get two possible solutions: $B = \frac{-7 + 47}{2} = \frac{40}{2} = 20$ or $B = \frac{-7 - 47}{2} = \frac{-54}{2} = -27 \quad (\text{Not a valid solution as breadth cannot be negative})$

So, $B = 20$ cm.

Step 4: Find the length

From Equation 2, we know: $L = B + 7 = 20 + 7 = 27 \, \text{cm}$

Step 5: Find the diagonal

The diagonal $D$ of a rectangle can be found using the Pythagorean theorem: $D = \sqrt{L^2 + B^2}$ Substitute $L = 27$ cm and $B = 20$ cm: $D = \sqrt{27^2 + 20^2} = \sqrt{729 + 400} = \sqrt{1129}$

Approximating: $D \approx 33.6 \, \text{cm}$

Final Answers:

• Length of the rectangle: 27 cm
• Breadth of the rectangle: 20 cm
• Diagonal of the rectangle: 33.6 cm (approximately)

Would you like more details or have any questions on this? Here are some related questions:

1. How would the area of the rectangle change if the length was reduced by a different amount?
2. How can the Pythagorean theorem be applied to other shapes?
3. What would happen to the diagonal if the breadth of the rectangle was increased?
4. How do you calculate the perimeter of a rectangle with given dimensions?
5. What are the properties of a square that makes it unique from a rectangle?

Tip: Always check the units when solving geometry problems—keeping track of units (like cm, m, etc.) ensures you're consistent and correct with your final answer.