The diagonals of a rhombus are of length 20 cm and 48 cm. What is the length of a side of the rhombus?
26 cm
Finding Rhombus Side Length from Diagonals
The question asks us to find the length of a side of a rhombus given the lengths of its diagonals. We are provided with the diagonal lengths as 20 cm and 48 cm.
A key property of a rhombus is that its diagonals bisect each other at right angles. This means that the point where the two diagonals intersect divides each diagonal into two equal parts, and the angle formed at the intersection is 90 degrees.
When the diagonals of a rhombus intersect, they divide the rhombus into four congruent right-angled triangles. The legs of each of these right triangles are half the lengths of the diagonals, and the hypotenuse of each right triangle is a side of the rhombus.
Let the lengths of the diagonals be \(d_1 = 20\) cm and \(d_2 = 48\) cm.
The half-lengths of the diagonals are:
- Half of diagonal 1 = \(\frac{d_1}{2} = \frac{20 \text{ cm}}{2} = 10\) cm
- Half of diagonal 2 = \(\frac{d_2}{2} = \frac{48 \text{ cm}}{2} = 24\) cm
These half-lengths are the lengths of the two legs of one of the right-angled triangles formed by the intersecting diagonals. The side of the rhombus is the hypotenuse of this right-angled triangle.
We can use the Pythagorean theorem to find the length of the hypotenuse (which is the side length 's' of the rhombus). The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (\(a^2 + b^2 = c^2\)).
Applying the Pythagorean theorem:
Let the legs be \(a = 10\) cm and \(b = 24\) cm, and the hypotenuse be \(s\) cm.
\(s^2 = a^2 + b^2\)
\(s^2 = (10 \text{ cm})^2 + (24 \text{ cm})^2\)
\(s^2 = 100 \text{ cm}^2 + 576 \text{ cm}^2\)
\(s^2 = 676 \text{ cm}^2\)
To find 's', we take the square root of both sides:
\(s = \sqrt{676 \text{ cm}^2}\)
\(s = 26 \text{ cm}\)
Therefore, the length of a side of the rhombus is 26 cm.
| Property | Value |
|---|---|
| Length of Diagonal 1 | 20 cm |
| Length of Diagonal 2 | 48 cm |
| Half of Diagonal 1 | 10 cm |
| Half of Diagonal 2 | 24 cm |
| Side of Rhombus (Hypotenuse) | 26 cm |
Rhombus Properties and Calculations
Understanding the properties of a rhombus is crucial for solving such problems. A rhombus is a quadrilateral with all four sides of equal length. Its diagonals have special properties:
- They bisect each other.
- They intersect at right angles.
- They bisect the angles of the rhombus.
These properties allow us to use the Pythagorean theorem when the diagonals and side lengths are involved.
Revision Table: Key Concepts
| Concept | Description | Relevance to Problem |
|---|---|---|
| Rhombus | A quadrilateral with four equal sides. | The shape in question. All sides have the length we are trying to find. |
| Diagonals of a Rhombus | Line segments connecting opposite vertices. | Given lengths (20 cm, 48 cm); they define the shape's internal structure. |
| Diagonal Bisection | Diagonals cut each other into two equal halves at their intersection. | Allows us to determine the legs of the right triangles. Half-diagonals are 10 cm and 24 cm. |
| Right Angle Intersection | Diagonals cross at a 90-degree angle. | Ensures the four triangles formed are right-angled, allowing the use of the Pythagorean theorem. |
| Pythagorean Theorem | \(a^2 + b^2 = c^2\) for a right triangle with legs a, b and hypotenuse c. | Used to calculate the side length (hypotenuse) from the half-diagonal lengths (legs). |
Additional Information: Related Concepts
Area of a Rhombus: The area of a rhombus can be calculated using its diagonals with the formula: Area \( = \frac{1}{2} \times d_1 \times d_2 \).
For this rhombus, the area would be \( \frac{1}{2} \times 20 \text{ cm} \times 48 \text{ cm} = 10 \text{ cm} \times 48 \text{ cm} = 480 \text{ cm}^2 \).
Rhombus vs. Square: A square is a special type of rhombus where all angles are 90 degrees. In a square, the diagonals are equal in length. In a non-square rhombus, the diagonals are unequal, as seen in this problem (20 cm and 48 cm).
Relation to Parallelogram: A rhombus is also a type of parallelogram (a quadrilateral with two pairs of parallel sides). However, a rhombus has the additional property of having all sides equal.
By understanding these properties and theorems like Pythagoras, various geometric problems involving rhombuses can be solved.
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