Question

1. a rhombus has diagonals of 6 and 8; find the length of a side and then the perimeter of the rhombus.

Explanation:

Step1: Recall rhombus property

The diagonals of a rhombus bisect each other at right - angles. Let the diagonals be \(d_1 = 6\) and \(d_2=8\). The half - lengths of the diagonals are \(a=\frac{d_1}{2}=\frac{6}{2} = 3\) and \(b=\frac{d_2}{2}=\frac{8}{2}=4\).

Step2: Use Pythagorean theorem to find side length

In one of the four right - triangles formed by the diagonals of the rhombus, if the side length of the rhombus is \(s\), then by the Pythagorean theorem \(s=\sqrt{a^{2}+b^{2}}\). Substituting \(a = 3\) and \(b = 4\), we get \(s=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\).

Step3: Calculate the perimeter

The perimeter \(P\) of a rhombus with side length \(s\) is given by \(P = 4s\). Since \(s = 5\), then \(P=4\times5=20\).

Answer:

The length of a side is 5 and the perimeter is 20.