Hello everyone. I’m working on the following problem involving a geometric progression and a quadrilateral:

Problem:

The lengths of the sides of a quadrilateral are in geometric progression and the longest side is 81 cm. Given that the perimeter is 120 cm, find the lengths of the other three sides.

Here’s what I’ve done with this problem so far:

Let the lengths of the four sides of the quadrilateral be denoted by t_1 < t_2 < t_3 < t_4.

t_4 = 81 cm (Given)

t_1 + t_2 + t_3 + t_4 = 120 cm (Given Perimeter of Quadrilateral)

Substituting for t_4 above:

t_1 + t_2 + t_3 + 81 = 120

t_1 + t_2 + t_3 = 39

a + ar + ar^2 = 39

a ( 1 + r + r^2 ) = 39

This tells us that the sum of the lengths of the other three sides equals 39 but we need to find the lengths for all three. How to proceed from here on...?? Need to basically find a and r.

Any and all help will be greatly appreciated. I’m a little rusty at this after many years of keeping away from math.

Thank you.

Shahz.