Let `A_(1)A_(2)A_(3)………………. A_(14)` be a regular polygon with 14 sides inscribed in a circle of radius 7 cm. Then the value of `(A_(1)A_(3))^(2) +(A_(1)A_(7))^(2) + (A_(3)A_(7))^(2)` (in square cm) is……………..

Explanation

To solve the problem, we need to find the value of \( (A_1A_3)^2 + (A_1A_7)^2 + (A_3A_7)^2 \) for a regular polygon with 14 sides inscribed in a circle of radius 7 cm.

### Step 1: Identify the lengths of the sides using the sine rule

For a regular polygon with \( n \) sides inscribed in a circle of radius \( R \), the length of the chord between two vertices \( A_i \) and \( A_j \) can be calculated using the formula:
\[ A_iA_j = 2R \sin\left(\frac{(j-i)\pi}{n}\right) \]
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