Question

This is a close-up of a larger tessellating pattern.The tessellation consists of two shapes. One of them is an equilateral triangle. What is the other?Regular hexagonO Regular octagonRegular decagonRegular dodecagon

Answer 1

Constructing the figure from the question

From the question, the triangle in the tessellation is an equilateral triangle

Hence, since equilateral triangle has all it angles to be equal then

`x=60^(\circ)`

From the tesselation,

`x+y+z=360^(\circ)`

But z = y

Therefore

`\begin{gathered} x+2y=360 \\ 60+2y=360 \\ 2y=360-60 \\ 2y=300 \\ y=(300)/(2) \\ y=150 \end{gathered}`

Hence one of the angle of the other shape of the tesselation is 150 degrees

Now we need to find the number of sides of the shape

applying number of sides of a polygon

Let n = number of sides

Then

`sum\text{ of interior angles =180(n-2)}`

But

sum of interior angles = n x 150 degrees

Therefore, we have

`150n=180(n-2)`

Now, we solve for n

`\begin{gathered} 150n=180n-360 \\ 360=180n-150n \\ 360=30n \\ n=(360)/(30) \\ n=12 \end{gathered}`

Hence the number of sides of the polygon is 12

A polygon having 12 sides is called dodecagon

Hence the answer is regular dodecagon