Rectangular Paper Dimensions and Area Calculation Using Folds

Suppose we have a rectangular paper, and when it is folded into two congruent parts along one set of sides, the perimeter of each part is 34 cm. When it is folded along the other set of sides, the perimeter for each part is 38 cm. This article will guide you through the steps to determine the dimensions of the paper and calculate its area.

Solution Process

Step 1: Define Variables and Understand the Folding Scenarios

Let's denote the length of the rectangular paper by l and the width by w.

Folding along one set of sides (length):

The dimensions of each part after folding become l/2 and w. The perimeter of each part is given by: P1 2(l/2 w) l 2w. We know that l 2w 34 cm.

Folding along the other set of sides (width):

The dimensions of each part after folding become l and w/2. The perimeter of each part is given by: P2 2(l w/2) 2l w. We know that 2l w 38 cm.

Step 2: Set Up the Equations

We now have two equations from our observations:

L 2w 34 2L w 38

Step 3: Solve the Equations

From the first equation: L 34 - 2w Substituting L into the second equation: 2(34 - 2w) w 38 68 - 4w w 38 68 - 3w 38 -3w 38 - 68 -3w -30 W 10 cm Substituting W back into the first equation: L 2(10) 34 L 20 34 L 14 cm

Step 4: Calculate the Area

NOW that we have the dimensions of the rectangle:

Length L 14 cm Width W 10 cm The area A of the rectangle is given by: A L u00d7 W 14 u00d7 10 140 cm2

Conclusion

The area of the paper is 140 cm2.

Alternative Solutions and Verification

Let the sheet of rectangular paper be L cm x B cm.

When folded along the midpoints of the length the perimeter 2BL 34 cm, or L 34–2B ...1

When folded along the midpoints of the breadth the perimeter B(2L) 38 cm ...2

Putting the value of L from 1 into 2

B(68 – 4B) 38 or 30 3B or B 10 cm. From 1 L 34 - 20 14 cm.

The area of the paper 14 u00d7 10 140 sq cm.

Further Reading

For a deeper understanding of similar problems involving areas and perimeters of rectangles, explore the following resources:

Additional Resources