Question

A regular m-gon has 3 times as many sides as a regular k-gon.

The measure of an interior angle of the regular m-gon is 14/3 times
the measure of an exterior angle of the regular k-gon.
Algebraically find the values of k and m.

Answer 1

9514 1404 393

Answer:

• k = 10
• m = 30

Explanation:

The interior angle of the m-gon is ...

m-interior = 180 -360/m

The exterior angle of the k-gon is ...

k-exterior = 360/k

The required relationships are ...

m = 3k

m-interior = 14/3(k-exterior)

Substituting for m, we can write the latter relation as ...

(180 -360/(3k)) = 14/3(360/k)

Multiplying by 3k/180, we have ...

3k -2 = 28

k = (28 +2)/3 = 10

The values of k and m are 10 and 30, respectively.

_____

Check

The interior angle of the m-gon is 180 -360/30 = 168 degrees.

The exterior angle of the k-gon is 360/10 = 36 degrees.

The angle ratio is 168/36 = 14/3 as required.