Understanding the Rhombus Problem

The question asks us to find the sum of the lengths of the diagonals of a rhombus, given the length of one side and the area. A rhombus is a special type of quadrilateral where all four sides are equal in length.

Key Properties of a Rhombus

To solve this problem, we need to recall some important properties of a rhombus:

• All four sides are equal in length. Let's denote the side length by \(s\).
• The diagonals bisect each other at right angles.
• The diagonals divide the rhombus into four congruent right-angled triangles.
• Let the lengths of the diagonals be \(d_1\) and \(d_2\). The semi-diagonals (\(\frac{d_1}{2}\) and \(\frac{d_2}{2}\)) are the legs of the right-angled triangles, and the side length \(s\) is the hypotenuse.

Formulas for a Rhombus

The relevant formulas for a rhombus are:

• Area (\(A\)): \(A = \frac{1}{2} \times d_1 \times d_2\)
• Relationship between side and diagonals (from Pythagorean theorem): \((\frac{d_1}{2})^2 + (\frac{d_2}{2})^2 = s^2\) which simplifies to \(d_1^2 + d_2^2 = 4s^2\).

Solving the Rhombus Diagonals Problem

We are given:

• Side length, \(s = 41\) cm
• Area, \(A = 720\) cm\(^2\)

We need to find the sum of the lengths of the diagonals, i.e., \(d_1 + d_2\).

Using the Area Formula

From the area formula, we have:

\(720 = \frac{1}{2} \times d_1 \times d_2\)

Multiplying both sides by 2, we get the product of the diagonals:

\(d_1 \times d_2 = 720 \times 2 = 1440\)

Using the Relationship between Side and Diagonals

From the Pythagorean relationship, we have:

\(d_1^2 + d_2^2 = 4s^2\)

Substitute the given side length \(s = 41\):

\(d_1^2 + d_2^2 = 4 \times (41)^2\)

Calculate \(41^2\):

\(41^2 = 1681\)

Now calculate \(4 \times 41^2\):

\(4 \times 1681 = 6724\)

So, we have:

\(d_1^2 + d_2^2 = 6724\)

Finding the Sum of the Diagonals

We need to find \(d_1 + d_2\). We know the algebraic identity:

\((d_1 + d_2)^2 = d_1^2 + d_2^2 + 2d_1 d_2\)

We have found the values for \(d_1^2 + d_2^2\) and \(d_1 d_2\):

• \(d_1^2 + d_2^2 = 6724\)
• \(d_1 d_2 = 1440\)

Substitute these values into the identity:

\((d_1 + d_2)^2 = 6724 + 2 \times 1440\)

Calculate \(2 \times 1440\):

\(2 \times 1440 = 2880\)

Now add the values:

\((d_1 + d_2)^2 = 6724 + 2880 = 9604\)

To find \(d_1 + d_2\), take the square root of 9604:

\(d_1 + d_2 = \sqrt{9604}\)

Let's find the square root of 9604. We can estimate that it's close to 100 (since \(100^2=10000\)) and must end in a digit whose square ends in 4 (which are 2 or 8). Let's try 98:

\(98 \times 98 = (100 - 2) \times (100 - 2) = 10000 - 200 - 200 + 4 = 10000 - 400 + 4 = 9604\)

So, the square root of 9604 is 98.

\(d_1 + d_2 = 98\)

The sum of the lengths of the diagonals is 98 cm.

Conclusion

Based on the calculations using the properties and formulas of a rhombus, the sum of the lengths of its diagonals is 98 cm.

Revision Table: Rhombus Diagonals and Area

Additional Information: Rhombus Geometry Concepts

Let's explore some related concepts about the rhombus geometry:

• Diagonals are Perpendicular Bisectors: The diagonals of a rhombus intersect at a right angle and cut each other exactly in half. This is a key property that allows us to use the Pythagorean theorem.
• Rhombus is a Parallelogram: A rhombus is a special type of parallelogram where all sides are equal. Therefore, it inherits all properties of a parallelogram, such as opposite angles being equal and consecutive angles being supplementary.
• Relationship with Squares: A square is a special case of a rhombus where all angles are 90 degrees. In a square, the diagonals are equal in length.
• Calculating Individual Diagonals: While the problem asks for the sum, you could potentially find the individual diagonal lengths by solving the system of equations: \(d_1 d_2 = 1440\) and \(d_1^2 + d_2^2 = 6724\). This would involve substitution and solving a quadratic equation, but finding the sum is more direct using the identity \((d_1 + d_2)^2\).