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Question
The interior angles of a polygon are in AP. The smallest angle is 120° and the common differences is 5". Find the number of sides of the polygon.
7
+8
9
10
ANSWER : 3
Descrption
<p style="margin-left:0px;">To find the number of sides of the polygon, you can use the following formula for the sum of interior angles in a polygon:</p><p style="margin-left:0px;">Sum of Interior Angles = (n - 2) × 180°</p><p style="margin-left:0px;">Where "n" is the number of sides of the polygon.</p><p style="margin-left:0px;">In this case, you are given that the smallest angle is 120°, and the common difference between the interior angles is 5°. So, you can set up an arithmetic progression (AP) to find the sum of the interior angles:</p><p style="margin-left:0px;">First angle (a) = 120° Common difference (d) = 5°</p><p style="margin-left:0px;">The sum of an AP is given by:</p><p style="margin-left:0px;">Sum = n/2 * [2a + (n-1)d]</p><p style="margin-left:0px;">Now, plug in the values:</p><p style="margin-left:0px;">Sum = n/2 * [2(120°) + (n-1)(5°)]</p><p style="margin-left:0px;">Simplify:</p><p style="margin-left:0px;">Sum = n/2 * [240° + 5°(n-1)]</p><p style="margin-left:0px;">Sum = n/2 * [240° + 5n - 5°]</p><p style="margin-left:0px;">Sum = n/2 * [5n + 235°]</p><p style="margin-left:0px;">Now, you know that the sum of interior angles of the polygon is equal to (n - 2) × 180°:</p><p style="margin-left:0px;">n/2 * [5n + 235°] = (n - 2) × 180°</p><p style="margin-left:0px;">Now, solve for "n":</p><p style="margin-left:0px;">n[5n + 235°] = 2(n - 2) × 180°</p><p style="margin-left:0px;">5n^2 + 235n = 2(180n - 360)</p><p style="margin-left:0px;">5n^2 + 235n = 360n - 720</p><p style="margin-left:0px;">Rearrange:</p><p style="margin-left:0px;">5n^2 - 125n - 720 = 0</p><p style="margin-left:0px;">Now, you can solve this quadratic equation for "n." You can either use the quadratic formula or factor it. Factoring, you get:</p><p style="margin-left:0px;">(n - 9)(5n + 80) = 0</p><p style="margin-left:0px;">Setting each factor equal to zero:</p><ol><li>n - 9 = 0 => n = 9</li><li>5n + 80 = 0 => 5n = -80 => n = -16 (but this doesn't make sense in this context)</li></ol><p style="margin-left:0px;">So, the number of sides of the polygon is 9.</p>
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