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Let $a_1 a_2 a_3 \dotsb a_{10}$ be a regular polygon. A rotation centered at $a_1$ with an angle of $\alpha$ takes $a_4$ to $a_8$. Given that $\alpha < 180^\circ$, find $\alpha,$ in degrees

Sagot :

sqdancefan

Answer:

  72°

Step-by-step explanation:

The required angle of rotation can be found by considering the regular polygon to be inscribed in a circle. Each of its 10 vertices will be separated by a central angle of 360°/10 = 36°. So, the vertices a8 and a4 will be separated by a central angle of (8 -4)(36°) = 144°.

The angle a4-a1-a8 is an inscribed angle, so will have half the measure of the central angle intercepting the a4-a8 arc. The desired angle of rotation to map a4 to a8 is ...

  α = 144°/2 = 72°

The value of angle α is 72° if the regular polygon is inscribed in a circle and rotation centered at a_1

What is a regular polygon?

A polygon is a geometric figure with a finite number of sides in two dimensions. A polygon's sides or edges are made up of straight-line segments that are joined end to end to form a closed shape. The vertices or corners are the points where two line segments meet, and an angle is generated as a result.

Consider the regular polygon to be inscribed in a circle to get the required angle of rotation.

A center angle of 360°

=360°/10 = 36°

It will separate each of its 10 vertices.

The central angle:

= (8 - 4)×(36°) = 144°

An inscribed angle is a4-a1-a8:

So α will be:

α = 144/2 ⇒ 72°

Thus, the value of angle α is 72° if the regular polygon is inscribed in a circle and rotation centered at a_1

Learn more about the regular polygon here:

https://brainly.com/question/11810316

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