# A side of a regular hexagon is three less than twice the length of a side of a regular decagon.If the perimeter of the hexagon equals the perimeter of decagon find a side of the hexagon.

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### 3 Answers

Let x and y be the sides of the regular hexagon and decagons respectively.

Then the perimeter of hexagon and decagons are 6x and 10y respectively. As the perimeters of both of them are said to be equal, we get the equation:

6x = 10y...........(1).

By second condition:

Side of the hexagon = 3 less than +2*side of decagon:

x = -3+2y..........(2)

Substituting x = -3+2y in (1):

6(-3+2y) = 10y

-18+12y = 10y

-18 = 10y -12y

-18 = -2y

-18/-2 = y.

y = 9.

Substituting y = 9 in the 2nd equation, we get: x = -3+2y.

x =-3+2*9 = 15.

Therefore the side of the hexagon = 15 and the side of the decagon = 9. Their perimeter = 15*6 = 9*10 = 90.

Let:

Length of a side of decagon = x

Then perimeter of decagon= 10*(Length of one side) = 10x

Also

Length of one side of hexagon = 2x - 3

And perimeter of hexagon = 6*(length of one side) = 6(2x - 3) = 12x - 18

Given perimeters of decagon and hexagons are equal:

Therefor:

10x = 12x - 18

==> 10x - 12x = -18

-2x = -18

x = 9

Side of hexagon = 2x - 3 = 2*9 - 3 = 15

Answer:

Side of hexagon = 15

The regular hexagon has 6 equal sides and the regular decagon has 10 equal sides.

We'll note as x = side of the regular decagon

According to the enunciation, the side of hexagon is 2x - 3.

We'll note the perimeter of decagon as P1 and the perimeter of hexagon as P2.

We'll also note the sides of the decagon as d1,d2... and the sides of the hexagon as h1,h2...

We'll calculate the perimeter of decagon:

P1 = d1 + d2 + ... + d10

Since d1 = d2 = ... = d10 = x

P1 = 10x

We'll calculate the perimeter of hexagon:

P2 = h1 + h2 + ... + h6

Since h1 = h2 = ... = h6 = 2x - 3

P2 = 6(2x - 3)

We know, from enunciation, that the perimeters are equals:

P1 = P2

10x = 6(2x-3)

We'll remove the brackets:

10x = 12x - 18

We'll subtract 12x - 18 both sides:

10x - 12x + 18 = 0

We'll combine like terms:

-2x + 18 = 0

We'll subtract 18 both sides:

-2x = -18

We'll divide by -2:

x = 9

**The side of the regular hexagon is:**

h = 2x - 3

h = 2*9 - 3

h = 18 - 3

**h = 15**