Question

If a polygon has 54 diagonal, then the number of sides are

Answer 1

Number of sides in the polygon are 12; having 54 diagonals.

Given:

• If a polygon has 54 diagonal.

To find:

• Find the number of sides.

Solution:

Formula to be used:

Relation between number of sides and number of diagonals is shown below:

If a polygon has 'n' sides and 'd' diagonals, then

$\bf \red{d = \frac{n(n - 3)}{2}}$

Step 1:

Write the given number of diagonals.

It is clearly given in the question,

d= 54

Step 2:

Find number of sides.

Put value of d in formula.

$54 = \frac{n(n - 3)}{2}$

or

$\bf {n}^{2} - 3n - 108 = 0$

solve the quadratic equation in n.

${n}^{2} - 12n + 9n - 108 = 0$

or

$n(n - 12) + 9(n - 12) = 0$

or

$(n + 9)(n - 12) = 0$

Find two values of n.

$n + 9 = 0$

or

$n = - 9$

we have to discard this value.

As number of sides in polygon are not negative.

or

$n - 12 = 0$

or

$\bf n = 12$

Thus,

Number of sides in the polygon are 12.

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Answer 2

Answer:

Sides of polygon is

Step-by-step explanation:

Given:

The diagonal of polygon = 54

To find:

Number of sides are

Calculations:

As we know that the number of polygon =$\frac{n(n-3)}{2}$

$\frac{n(n-3)}{2}=54$

$n(n-3)=$$54$×$2$

$n^{2} -3n=108$

$n^{2}- 3n-108=0\\n^{2} -12n+9n-108=0\\n(n-12)+9(n-12)=0\\(n-12)(n+9)=0\\(n-12)=0\\n=12\\(n+9)=0\\n=-9$

Sides cannot be negative.

Therefore,$n=12$

Hence, sides of polygon is 12.