# Find the number of sides of each of the two polygons if the total number of sides of the polygons is 13, and the sum of the number of diagonals of the

polygon is 25.

1
by rikatadeja 29.10.2015

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polygon is 25.

1
by rikatadeja 29.10.2015

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X + y = 13

y = 13 - x

Representation:

*x* = total number of *sides of ***first polygon**

__13 - x __= total number of* sides of ***second polygon.**

*Number of diagonals in a polygon:*

where n = number of sides of a polygon

Number of diagonals in__first polygon__ with __x sides__.

Substitute x for n:

=__x (x-3)__

2

=*x² - 3x*

**2**

Number of diagonals in__second polygon__ with __13-x sides.__

Substitute 13 - x for n:

=__13 - x (13 - x - 3)__

2

=__13 - x (10 - x)__

2

=__130 -13x - 10x + x²__

2

=__ __**x² - 23x + 130**

** 2**

*The sum of diagonals of the two polygons is 25.*

__x____² - 3x __+ __x² - 23x + 130 __= 25

2 2

__x² + x² -3x - 23x + 130__ = 25

2

2 (__2x² - 26x + 130__ = 25) 2

2

2x² - 26x + 130 = 50

2x² - 26x + 130 - 50 = 0

2x² - 26x + 80 = 0

Factor out 2 (GCF)

2 (x² - 13x + 40) = 0

Factor the quadratic equation, then solve for the roots (x):

x² - 13x + 40 = 0

(x - 8) (x - 5) = 0

x - 8 = 0 x - 5 = 0

__x = 8 __ ** x = 5**

The number of sides of each polygon:

First polygon:x ⇒ x =**8 sides **(octagon)

Second polygon: 13 - x ⇒ 13 - 8 =**5 sides **(pentagon)

To check if 8 and 5 sides are correct:

The sum of the sides of the two polygons is 13

8 + 5 = 13

13 = 13

The sum of the diagonals of the two polygons is 25:

**Diagonals of first polygon with 8 sides**, where n = sides:

__n (n-3)__ =__ 8 (8-3)__ = __8(5)__ = __40 __ = **20 diagonals**

2 2 2 2

**Diagonals of the second polygon with 5 sides**, where n = sides:

__n(n-3)__ = __5 (5-3) __ = __5(2)__ = __10 __ = **5 diagonals**

2 2 2 2

Add the diagonals:

20 + 5 = 25

25 = 25

__FINAL ANSWER: The number of sides of the two polygons are 8 and 5 sides.__

y = 13 - x

Representation:

where n = number of sides of a polygon

Number of diagonals in

Substitute x for n:

=

2

=

Number of diagonals in

Substitute 13 - x for n:

=

2

=

2

=

2

=

2 2

2

2 (

2

2x² - 26x + 130 = 50

2x² - 26x + 130 - 50 = 0

2x² - 26x + 80 = 0

Factor out 2 (GCF)

2 (x² - 13x + 40) = 0

Factor the quadratic equation, then solve for the roots (x):

x² - 13x + 40 = 0

(x - 8) (x - 5) = 0

x - 8 = 0 x - 5 = 0

The number of sides of each polygon:

First polygon:x ⇒ x =

Second polygon: 13 - x ⇒ 13 - 8 =

To check if 8 and 5 sides are correct:

The sum of the sides of the two polygons is 13

8 + 5 = 13

13 = 13

The sum of the diagonals of the two polygons is 25:

2 2 2 2

2 2 2 2

Add the diagonals:

20 + 5 = 25

25 = 25