Question

What is the minimum angle of rotation (in degrees) that will carry the star onto itself?

Answer 1

Answer:

Minimum angle = 108°

Step-by-step explanation:

From the figure attached,

Minimum angle to overlap the star onto itself is the angle of rotation from point A to B.

Let the angle between A and B measure a°.

If we complete a star pentagon by joining the vertices of the star,

Formula to measure the internal angle of a polygon is,

Measure of internal angle = $180\times \frac{(n-2)}{n}$

Measure of ∠ABC = $\frac{(5-2)}{5}\times 180$

                              = 108°

Since, one side star divides the interior angle into three equal parts,

Measure of interior angle = $\frac{108}{3}$ = 36°

From ΔAOB,

m(∠OAB) + m(∠AOB) + m(∠ABO) = 180° [Sum of interior angles of a triangle = 180°]

36° + a + 36° = 180°

a + 72° = 180°

a = 180 - 72

a = 108°

Therefore, by the minimum angle of rotation of 108° star will overlap onto itself.