Question

My first Question


- A convex polygon has 44 diagnols. Find the number of its sides.

Answer 1

Heya !
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Ques.→ A convex polygon has 44 Diagnols. Find the number of its sides .

Ans.→ Let us assume that the convex polygon has n number of sides .

•°• We have ,

=> No. Of Diagnols + No. Of Sides = ⁿ C 2

=> 44 + No. Of sides = n! / 2! ( n-2)! [ as nCr = n!/r! ( n-r)! ]

$= > 44 + n = \frac{n(n - 1)(n - 2)}{2 \times 1(n - 2)}$

[ inserting the values and opening the factorials. ]

•( n - 2 )! Gets cancelled and we have ,

→ 2 ( 44 + n ) = n² - n

→ 88 + 2n = n² - n

→ n² - 3n - 88 = 0

=> n² + 8n - 11 n - 88 = 0 [ Using Middle Term splitting ]

=> n ( n + 8 ) -11 ( n + 8)

=> ( n - 11 ) ( n + 8)

•°• n = -8 , n = 11 ✔

★Number of sides can not be negative . So n = 11 ✔

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Answer 2

$\boxed{Ans => 11}$

Question ➡

A convex polygon has 44 diagnals. Find the number of its sides.

Answer ➡

Let the number of sides of polygon =n

Number of angular points =n

Number of straight lines joining any two of these n points =nC2nC2

Now the number of sides of the polygon =n

Number of diagonals =nC2−nnC2−n

But it is given the number of diagonals =44

nC2−n=44

But it is given the number of diagonals =44

n(n−1)/2-n=44

n2−n−2n=88

n2−3n−88=0

(n−11)(n+8)=0

n=11 or n=−8

Rejecting negative quantity,

n=11

Hence the required number of sides = 11