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Let A(1),A(2),A(3),...A(12) are vertices...

Let `A_(1),A_(2),A_(3),...A_(12)` are vertices of a regular dodecagon. If radius of its circumcircle is 1, then the length `A_(1)A_(3)` is-

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To find the length of the segment \( A_1A_3 \) in a regular dodecagon (12-sided polygon) inscribed in a circle of radius 1, we can follow these steps: ### Step 1: Understand the Geometry A regular dodecagon has 12 vertices equally spaced around a circle. The angle between any two adjacent vertices is \( \frac{360^\circ}{12} = 30^\circ \). ### Step 2: Determine the Angles The vertices \( A_1 \) and \( A_3 \) correspond to angles at the center of the circle. The angle \( A_1OA_3 \) (where \( O \) is the center of the circle) can be calculated as: \[ \text{Angle} = 2 \times 30^\circ = 60^\circ \] ### Step 3: Use the Law of Cosines To find the length \( A_1A_3 \), we can use the Law of Cosines in triangle \( A_1OA_3 \): \[ A_1A_3^2 = A_1O^2 + A_3O^2 - 2 \cdot A_1O \cdot A_3O \cdot \cos(60^\circ) \] Since both \( A_1O \) and \( A_3O \) are equal to the radius of the circumcircle, which is 1: \[ A_1A_3^2 = 1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \cos(60^\circ) \] \[ A_1A_3^2 = 1 + 1 - 2 \cdot \frac{1}{2} \] \[ A_1A_3^2 = 2 - 1 = 1 \] \[ A_1A_3 = \sqrt{1} = 1 \] ### Conclusion The length \( A_1A_3 \) is \( 1 \) unit. ---
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