Question

Please help with this question. n refers to the number of sides.

Answer 1

n = 18

Explanation:

The sides in a regular polygon are equal in length.

⇒ BC = CD

If BC = CD then ΔDCB is an isosceles triangle.

As the base angles of an isosceles triangles are equal:

⇒ ∠DBC = ∠BDC = 10°

Interior angles in a triangle sum to 180°.

⇒ ∠DBC +∠BDC + ∠DCB = 180°

⇒ 10° + 10° + ∠DCB = 180°

⇒ 20° + ∠DCB = 180°

⇒ ∠DCB = 180° - 20°

⇒ ∠DCB = 160°

Therefore, the interior angle of the regular polygon is 160°.

The interior angles of a regular polygon are equal in size.

`\boxed{\begin{minipage}{5.2 cm}\underline{Interior angle of a polygon}\\\\$((n-2) * 180^(\circ))/(n)$\\\\where $n$ is the number of sides.\\ \end{minipage}}`

To find the number of sides, equate the formula to the found value of the interior angle and solve for n:

`\begin{aligned}\implies((n-2) * 180^(\circ))/(n) & =160^(\circ)\\(n-2) * 180^(\circ) & =n * 160^(\circ)\\ 180(n-2) & = 160n\\180n-360 & = 160n\\180n & = 160n+360\\180n-160n & = 360\\20n & = 360\\n &= (360)/(20)\\n & =18\\\end{aligned}`

Therefore, the regular polygon has 18 sides.