Find the length of a side of a rhombus whose diagonals are 6cm and 8cm

Analysis

To find the side length of a rhombus when the diagonals are given, we must use the geometric properties of a rhombus. A rhombus has diagonals that bisect each other at right angles ($90^\circ$).

Concept

When the diagonals of a rhombus intersect, they divide the rhombus into four congruent right-angled triangles.

• The diagonals are $d_1 = 6\text{ cm}$ and $d_2 = 8\text{ cm}$.
• Because they bisect each other, the legs of each right-angled triangle are half the lengths of the diagonals: • Leg 1 = $\frac{d_1}{2} = \frac{6}{2} = 3\text{ cm}$ • Leg 2 = $\frac{d_2}{2} = \frac{8}{2} = 4\text{ cm}$
• The side of the rhombus ($s$) acts as the hypotenuse of these right-angled triangles. We can use the Pythagorean theorem: $a^2 + b^2 = c^2$.

Solution

1. Identify the legs of the triangle: • $a = 3\text{ cm}$ • $b = 4\text{ cm}$

2. Apply the Pythagorean theorem: • $s^2 = 3^2 + 4^2$ • $s^2 = 9 + 16$ • $s^2 = 25$

3. Solve for $s$: • $s = \sqrt{25}$ • $s = 5\text{ cm}$

The length of the side of the rhombus is 5cm.

Exam Tip

Whenever you see a rhombus problem involving diagonals of 6 and 8, remember the 3-4-5 Pythagorean triple. It is a very common pattern in JAMB Mathematics questions, and recognizing it can save you precious time during the exam!