Perimeter of a rhombus

Question

Find the perimeter of a rhombus whose diagonals measure 14cm and 18cm. Express in simplest radical form.

Answer 1

The nice thing about rhombi vs general parallelograms is that the two diagonals are perpendicular to each other! They also bisect (cut in half) each other, which means we can draw a right triangle whose two legs are half of each diagonal and whose hypotenuse is a side of the rhombus. Call the side length s. Then by the pythagorean theorem we have

s^2 = (14/2)^2+(18/2)^2

= 7^2+9^2

= 49+81

= 130

This means that s = sqrt(130). But each side of a rhombus is the same length, so the perimeter is 4s, so

4sqrt(130). This is already in simplest radical form because 130 = 2*5*13 is not divisible by any squares.

Answer 2

The product of prime numbers to the first power cannot be factored by any squares, and hence is an irreducible radical.