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Home » A convex polygon has 44 diagonals. Find the number of its sides. [Hint: Polygon of sides has ) number of diagonals.]
A convex polygon has 44 diagonals. Find the number of its sides. [Hint: Polygon of n sides has \left({ }^{n} C_{2}\right) -\mathrm{n} ) number of diagonals.]
CBSE | Class 11 | Maths | NCERT Exemplar | Permutations and Combinations
A convex polygon has 44 diagonals. Find the number of its sides. [Hint: Polygon of n sides has \left({ }^{n} C_{2}\right) -\mathrm{n} ) number of diagonals.]

Solution:

It is known that,

\begin{array}{l} { }^{n} C_{r} \\ =\frac{n !}{r !(n-r) !} \end{array}

Let’s suppose the no. of sides the given polygon have =\mathrm{n}

So now,

The no. of line segments obtained by joining n vertices ={ }^{n} C_{2}

Therefore, the no. of diagonals of the polygon ={ }^{\mathrm{n}} \mathrm{C}_{2}-\mathrm{n}=44

\begin{array}{l} \frac{n(n-1)}{2}-n=44 \\ n^{2}-3 n-88=0 \\ (n-11)(n+8)=0 \\ n=11 \text { or } n=-8 \end{array}

As a result, the polygon has 11 sides.

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