Understanding the Relationship Between Diagonal and Perimeter

Oct 13, 2025 by windailys.com 64 views

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Introduction

In this article, we will explore the relationship between the diagonal and perimeter of a rectangle to find its area. The given information includes the diagonal of the rectangle, which is 15cm, and the perimeter of the rectangle, which is 42cm. We will use these values to calculate the area of the rectangle.

Understanding the Relationship Between Diagonal and Perimeter

The diagonal of a rectangle divides it into two right-angled triangles. The relationship between the diagonal and the sides of the rectangle can be expressed using the Pythagorean theorem. However, in this case, we are given the perimeter of the rectangle, which is the sum of all its sides. We can use this information to find the length and width of the rectangle.

Calculating the Length and Width of the Rectangle

Let's denote the length of the rectangle as 'l' and the width as 'w'. The perimeter of the rectangle is given by the formula:

P = 2(l + w)

We are given that the perimeter of the rectangle is 42cm. Substituting this value into the formula, we get:

42 = 2(l + w)

Simplifying the equation, we get:

l + w = 21

Using the Diagonal to Find the Length and Width

We are also given that the diagonal of the rectangle is 15cm. Using the Pythagorean theorem, we can express the relationship between the diagonal, length, and width as:

d^2 = l^2 + w^2

where d is the diagonal. Substituting the given value of the diagonal, we get:

15^2 = l^2 + w^2

Simplifying the equation, we get:

225 = l^2 + w^2

Solving the System of Equations

We have two equations:

l + w = 21 l^2 + w^2 = 225

We can solve this system of equations to find the values of 'l' and 'w'. One way to do this is to express 'w' in terms of 'l' using the first equation and substitute it into the second equation.

From the first equation, we get:

w = 21 - l

Substituting this expression into the second equation, we get:

l^2 + (21 - l)^2 = 225

Expanding and simplifying the equation, we get:

l^2 + 441 - 42l + l^2 = 225

Combine like terms:

2l^2 - 42l + 216 = 0

Divide the equation by 2:

l^2 - 21l + 108 = 0

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

l = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -21, and c = 108. Substituting these values into the formula, we get:

l = (21 ± √((-21)^2 - 4(1)(108))) / 2(1)

Simplifying the equation, we get:

l = (21 ± √(441 - 432)) / 2

l = (21 ± √9) / 2

l = (21 ± 3) / 2

We have two possible values for 'l':

l = (21 + 3) / 2 = 12 l = (21 - 3) / 2 = 9

Finding the Width

Now that we have the values of 'l', we can find the corresponding values of 'w' using the equation:

w = 21 - l

Substituting the values of 'l', we get:

w = 21 - 12 = 9 w = 21 - 9 = 12

Calculating the Area of the Rectangle

Now that we have the values of 'l' and 'w', we can calculate the area of the rectangle using the formula:

A = l × w

Substituting the values of 'l' and 'w', we get:

A = 12 × 9 = 108 A = 9 × 12 = 108

We have two possible values for the area of the rectangle, but they are the same. This is because the length and width of the rectangle are interchangeable.

Conclusion

In this article, we used the given information about the diagonal and perimeter of a rectangle to find its area. We first used the perimeter to find the length and width of the rectangle, and then used the diagonal to solve for the length and width. We found that the area of the rectangle is 108 square centimeters.

Frequently Asked Questions

Q: What is the relationship between the diagonal and perimeter of a rectangle? A: The diagonal of a rectangle divides it into two right-angled triangles. The relationship between the diagonal and the sides of the rectangle can be expressed using the Pythagorean theorem.
Q: How can we use the perimeter to find the length and width of a rectangle? A: We can use the formula P = 2(l + w) to find the length and width of a rectangle.
Q: How can we use the diagonal to find the length and width of a rectangle? A: We can use the Pythagorean theorem to express the relationship between the diagonal, length, and width as d^2 = l^2 + w^2.

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Q&A: Diagonal of the Rectangle is 15cm and Perimeter of Rectangle is 42cm

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Q: What is the relationship between the diagonal and perimeter of a rectangle?

A: The diagonal of a rectangle divides it into two right-angled triangles. The relationship between the diagonal and the sides of the rectangle can be expressed using the Pythagorean theorem.

Q: How can we use the perimeter to find the length and width of a rectangle?

A: We can use the formula P = 2(l + w) to find the length and width of a rectangle. Given the perimeter of the rectangle, we can set up an equation to solve for the length and width.

Q: How can we use the diagonal to find the length and width of a rectangle?

A: We can use the Pythagorean theorem to express the relationship between the diagonal, length, and width as d^2 = l^2 + w^2. This equation can be used to solve for the length and width of the rectangle.

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is A = l × w, where l is the length and w is the width.

Q: How can we find the area of a rectangle when given the diagonal and perimeter?

A: To find the area of a rectangle when given the diagonal and perimeter, we can follow these steps:

Use the perimeter to find the length and width of the rectangle.
Use the diagonal to solve for the length and width of the rectangle.
Calculate the area of the rectangle using the formula A = l × w.

Q: What is the area of the rectangle when the diagonal is 15cm and the perimeter is 42cm?

A: The area of the rectangle is 108 square centimeters.

Q: Can we have multiple solutions for the length and width of the rectangle?

A: Yes, we can have multiple solutions for the length and width of the rectangle. This is because the length and width are interchangeable.

Q: How can we determine which solution is correct?

A: To determine which solution is correct, we can use the given information to check if the length and width satisfy the conditions. In this case, we can check if the length and width satisfy the equation l + w = 21.

Q: What is the significance of the Pythagorean theorem in this problem?

A: The Pythagorean theorem is used to express the relationship between the diagonal, length, and width of the rectangle. It is a fundamental concept in geometry and is used to solve problems involving right-angled triangles.

Q: Can we use other formulas to find the area of the rectangle?

A: Yes, we can use other formulas to find the area of the rectangle. For example, we can use the formula A = (d^2) / 2, where d is the diagonal.

Q: What is the relationship between the diagonal and the area of the rectangle?

A: The diagonal and the area of the rectangle are related through the Pythagorean theorem. The area of the rectangle can be expressed in terms of the diagonal and the length and width of the rectangle.

Q: Can we find the length and width of the rectangle using only the diagonal?

A: No, we cannot find the length and width of the rectangle using only the diagonal. We need additional information, such as the perimeter, to solve for the length and width.

Q: What is the significance of the perimeter in this problem?

A: The perimeter is used to find the length and width of the rectangle. It is a fundamental concept in geometry and is used to solve problems involving rectangles and other polygons.

Q: Can we use the perimeter to find the area of the rectangle?

A: Yes, we can use the perimeter to find the area of the rectangle. We can use the formula A = (P^2) / 4, where P is the perimeter.

Q: What is the relationship between the perimeter and the area of the rectangle?

A: The perimeter and the area of the rectangle are related through the formula A = (P^2) / 4. This formula can be used to find the area of the rectangle when the perimeter is given.