Peter has a set of polygonal tiles, where all the polygons are regular and have the same side length. He find that two pentagons and a decagon can fit together perfectly. Peter also find that a triangle, an octagon, and an n-gon also fit together perfectly. Find n. (Remember that all the tiles are regular polygons.)

24 sides

Step-by-step explanation:

Angle at a point = 360°

To find the number of degrees in each interior angle of a regular polygon, we make use of

180(n - 2)/n

There are 8 sides in a regular octagon,

Each interior of a regular octagon= 180(8-2)/8

= 6 * 180/8

= ¾ * 180

= 135°

The interior angle of a regular triangle (equilateral triangle) is 60°

The interior angle of the regular n-gon is given by

360° - 135 - 60°

= 165°

Since the interior angle is 165°

Then

180(n - 2)/n = 165 for the regular n-gon

Solving for n

Multiply both sides by n

180(n - 2) = 165n Open the bracket

180n - 360 = 165n Collect Like Terms

180n - 165n = 360

15n = 360 Divide both sides by 15

n = 360/15

n = 24

Thus, the number of sides of the regular n-gon is 24