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Difficulty: MediumAngles, Parallel Lines, and Polygons

The ratio of the measure of each interior angle to each exterior angle of a regular convex polygon is 7:27 : 2. What is the total number of diagonals of this polygon?

  1. 27Answer
  2. B
    9
  3. C
    36
  4. D
    54

Answer

The total number of diagonals of the polygon is 27.
Since interior and exterior angles are supplementary, their ratio 7:27 : 2 divides 180∘180^\circ into 9 equal parts of 20∘20^\circ. The exterior angle is 2×20∘=40∘2 \times 20^\circ = 40^\circ. The number of sides nn is 360∘/40∘=9360^\circ / 40^\circ = 9. Using the formula for diagonals D=n(n−3)2D = \frac{n(n-3)}{2}, we obtain D=9×62=27D = \frac{9 \times 6}{2} = 27.

Step-by-Step Solution

1
Calculate the measure of the exterior angle
Exterior angle =27+2×180∘=40∘= \frac{2}{7 + 2} \times 180^\circ = 40^\circ
Interior and exterior angles at a vertex are supplementary, summing to 180∘180^\circ.
2
Determine the number of sides nn
n=360∘40∘=9n = \frac{360^\circ}{40^\circ} = 9
The sum of exterior angles of any convex polygon is 360∘360^\circ, so n=360∘exterior anglen = \frac{360^\circ}{\text{exterior angle}}.
3
Calculate the number of diagonals using D=n(n−3)2D = \frac{n(n - 3)}{2}
D=9(9−3)2=9×62=27D = \frac{9(9 - 3)}{2} = \frac{9 \times 6}{2} = 27
Each vertex connects to n−3n - 3 non-adjacent vertices, and dividing by 2 avoids double-counting.

Key Concept

Interior and exterior angles of regular polygons and the polygon diagonal count formula.
Estimated Time:1m 30s
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Topics

Angles, Parallel Lines, and Polygons

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