Mathematics Times – July 2019

in the n-gon. An n-gon will have exterior angles

of

360

n

(the sum of all the exterior angles must

add up to 360 degrees). Consider the pentagon

and hexagon:

Therefore, the interior angles of the hexagon

are 120 degrees each. In an edge to edge tiling,

at a point where the corners of hexagons meet,

we would require the interior angles to sum to

360 degrees. This happens if we have 3

hexagons, and we see that our regular tiling

indeed has three hexagons meeting at each

corner next line.

The interior angles of the pentagon are 108

degrees each. At a point where the corners of

pentagons meet, we would require the interior

angles to sum to 360 degrees. Three pentagons

would be 324 degrees, which is not enough,

and four pentagons would be 432 degrees,

which is too much. This is why we cannot tile

the plane with pentagons.

This also explains why we have four squares
meeting at each corner for squares
(4 90 360)  , and six triangles meeting at
each corner for triangles (6 60 360) .
The only edge to edge regular tilings are the
ones with equilateral triangles, squares, or
hexagons.
5.Sol: We can solve this problem in two methods,
one is using graph and the other is using
algebra.

Graphical Method

From the graph, we can observe there are

four intersecting points, that is four points

satisfying the simultaneous equations.

algebraic Method Given simultaneous

equations are rewritten as

x y ^212

x y^2   12

Which yields x y x y^2   ^2. That is

x y x y^2   ^2. After simplifying, we get

x y  0 and x y  1

Now

x = y: Implies that x x^2   12

i.e., x x^2   12 0

Solving the above equation we get x=-4,3.

Therefore corresponding y coordinates are -

4, 3 respectively.

Thus two intersecting are ( 4, 4)  and (3,3).

x + y = 1: That isy  1 x. Substituting in

any of the equation and solving for

5.Sol:

Graphical Method

1 1 44

2

x

 

. we can get two

corresponding values y by substituting x

values. Hence it has two intersecting points.