Finding the Area of a Rhombus

The question asks us to calculate the area of a rhombus given its perimeter and the length of one diagonal. We know that a rhombus is a quadrilateral with all four sides of equal length. Its diagonals bisect each other at right angles.

Understanding Rhombus Properties

Key properties of a rhombus useful for solving this problem:

• All sides are equal.
• Diagonals bisect each other perpendicularly.
• The two diagonals divide the rhombus into four congruent right-angled triangles.
• The area of a rhombus can be calculated using the formula: Area = \(\frac{1}{2} \times d_1 \times d_2\), where \(d_1\) and \(d_2\) are the lengths of the diagonals.

Step-by-Step Calculation

1. Find the side length of the rhombus

The perimeter of a rhombus is the sum of the lengths of its four equal sides. If the perimeter is 116 cm, we can find the length of one side (s) by dividing the perimeter by 4.

Perimeter = \(4 \times \text{side}\)

\(116 \text{ cm} = 4 \times s\)

\(s = \frac{116}{4} \text{ cm}\)

\(s = 29 \text{ cm}\)

So, the length of each side of the rhombus is 29 cm.

2. Determine the half-lengths of the diagonals

We are given that one diagonal (\(d_1\)) is 42 cm long. Since the diagonals of a rhombus bisect each other, half of this diagonal will be:

Half diagonal 1 = \(\frac{d_1}{2} = \frac{42}{2} \text{ cm} = 21 \text{ cm}\)

Let the other diagonal be \(d_2\). Half of the other diagonal is \(\frac{d_2}{2}\).

3. Use the Pythagorean Theorem to find the length of the other diagonal

The diagonals of a rhombus intersect at right angles. This means that half of each diagonal and one side of the rhombus form a right-angled triangle. The side of the rhombus is the hypotenuse of this triangle. We can use the Pythagorean theorem (\(a^2 + b^2 = c^2\)) where 'a' and 'b' are the legs (half-diagonals) and 'c' is the hypotenuse (side).

\((\text{Half diagonal 1})^2 + (\text{Half diagonal 2})^2 = (\text{Side})^2\)

\(\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = s^2\)

Substitute the known values:

\((21)^2 + \left(\frac{d_2}{2}\right)^2 = (29)^2\)

\(441 + \left(\frac{d_2}{2}\right)^2 = 841\)

Now, solve for \(\left(\frac{d_2}{2}\right)^2\):

\(\left(\frac{d_2}{2}\right)^2 = 841 - 441\)

\(\left(\frac{d_2}{2}\right)^2 = 400\)

Take the square root of both sides to find \(\frac{d_2}{2}\):

\(\frac{d_2}{2} = \sqrt{400}\)

\(\frac{d_2}{2} = 20 \text{ cm}\)

Now, find the length of the full diagonal \(d_2\):

\(d_2 = 2 \times 20 \text{ cm}\)

\(d_2 = 40 \text{ cm}\)

The length of the other diagonal is 40 cm.

4. Calculate the area of the rhombus

Now that we have the lengths of both diagonals (\(d_1 = 42 \text{ cm}\) and \(d_2 = 40 \text{ cm}\)), we can calculate the area using the formula:

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

Area = \(\frac{1}{2} \times 42 \text{ cm} \times 40 \text{ cm}\)

Area = \(21 \times 40 \text{ cm}^2\)

Area = \(840 \text{ cm}^2\)

Summary of Results

The calculated area of the rhombus is 840 cm\(^2\).

Revision Table: Rhombus Properties and Formulas

Additional Information: Understanding Rhombus Geometry

A rhombus is a special type of parallelogram where all sides are equal. While all rhombuses are parallelograms, not all parallelograms are rhombuses (unless their sides are equal). A square is a special type of rhombus where all angles are right angles, which means its diagonals are also equal in length.

The property that diagonals of a rhombus are perpendicular bisectors of each other is crucial for finding unknown lengths using the Pythagorean theorem, as demonstrated in this problem. The four right-angled triangles formed by the intersecting diagonals within the rhombus are congruent, which helps in understanding the symmetry and properties of the shape.