The question in the image is in Gujarati and translates to:

Question: If the perimeter of a rectangle is 80 meters and the area is 400 square meters, can such a rectangle be constructed? If possible, find its length and breadth.

Let's solve this step by step:

Given:

• Perimeter $P = 80$ meters
• Area $A = 400$ square meters

Formulae:

• The perimeter of a rectangle:
  $P = 2(l + b)$ where $l$ is the length and $b$ is the breadth.

• The area of a rectangle:
  $A = l \times b$

Step 1: Solve for the perimeter

From the perimeter equation $P = 2(l + b)$, we have: $80 = 2(l + b)$ $l + b = 40 \quad \text{(Equation 1)}$

Step 2: Solve for the area

The area equation gives us: $l \times b = 400 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

From Equation 1, we can express $l$ as: $l = 40 - b$ Substitute this into Equation 2: $(40 - b) \times b = 400$ $40b - b^2 = 400$ Rearranging this equation: $b^2 - 40b + 400 = 0$

This is a quadratic equation, which can be solved using the quadratic formula: $b = \frac{-(-40) \pm \sqrt{(-40)^2 - 4(1)(400)}}{2(1)}$ $b = \frac{40 \pm \sqrt{1600 - 1600}}{2}$ $b = \frac{40 \pm \sqrt{0}}{2}$ $b = \frac{40}{2} = 20$

Step 4: Find the length

Using $b = 20$ in Equation 1: $l + 20 = 40$ $l = 40 - 20 = 20$

Step 5: Conclusion

Both the length and breadth of the rectangle are 20 meters, meaning the rectangle is a square.

Thus, the rectangle can be constructed, and its length and breadth are both 20 meters.

Would you like more details or further clarifications?

Here are five related questions:

1. What is the difference between the perimeter and area of a rectangle?
2. Can a rectangle always be a square if its length equals its breadth?
3. How do you solve quadratic equations in geometry problems?
4. What are other ways to find the dimensions of a rectangle given the perimeter and area?
5. How do the dimensions of a rectangle change when the perimeter is constant but the area changes?

Tip: Always check for consistency between the perimeter and area equations when solving geometry problems involving rectangles.