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Difficulty: HardQuadrilaterals and Polygons

The ratio of the measure of each interior angle of a regular nn-sided polygon PP to the measure of each interior angle of a regular (n+2)(n+2)-sided polygon is 2425\frac{24}{25}. Which of the following statements about polygon PP must be true? Select all such statements.

  1. Polygon PP has 1010 sides.Answer
  2. Polygon PP has 3535 diagonals.Answer
  3. C
    The measure of each exterior angle of polygon PP is 40∘40^\circ.
  4. D
    The sum of the measures of the interior angles of polygon PP is 1800∘1{}800^\circ.
  5. The measure of each interior angle of polygon PP is 144∘144^\circ.Answer

Answer

The statements asserting that polygon P has 10 sides, polygon P has 35 diagonals, and each interior angle of polygon P measures 144 degrees are all correct.
To determine which statements are true, we set up the ratio of the interior angle of a regular nn-sided polygon to that of a regular (n+2)(n+2)-sided polygon: (n−2)(n+2)n2=2425\frac{(n-2)(n+2)}{n^2} = \frac{24}{25}. Simplifying gives 1−4n2=24251 - \frac{4}{n^2} = \frac{24}{25}, which leads to n2=100n^2 = 100, so n=10n = 10. Therefore, polygon PP is a regular decagon (10 sides). Evaluating the properties of a regular 10-gon shows that the number of diagonals is 10(10−3)2=35\frac{10(10-3)}{2} = 35, each interior angle measures 8⋅180∘10=144∘\frac{8 \cdot 180^\circ}{10} = 144^\circ, each exterior angle measures 360∘10=36∘\frac{360^\circ}{10} = 36^\circ, and the sum of the interior angles is 8⋅180∘=1440∘8 \cdot 180^\circ = 1{}440^\circ. Consequently, the options stating that the polygon has 10 sides, has 35 diagonals, and has interior angles measuring 144∘144^\circ are correct.

Step-by-Step Solution

1
Set up the algebraic equation comparing the interior angles of an nn-gon and an (n+2)(n+2)-gon.
(n−2)⋅180∘nn⋅180∘n+2=2425  ⟹  (n−2)(n+2)n2=2425\frac{\frac{(n-2) \cdot 180^\circ}{n}}{\frac{n \cdot 180^\circ}{n+2}} = \frac{24}{25} \implies \frac{(n-2)(n+2)}{n^2} = \frac{24}{25}
The formula for each interior angle of a regular polygon with kk sides is (k−2)⋅180∘k\frac{(k-2) \cdot 180^\circ}{k}.
2
Solve for nn.
1−4n2=2425  ⟹  4n2=125  ⟹  n2=100  ⟹  n=101 - \frac{4}{n^2} = \frac{24}{25} \implies \frac{4}{n^2} = \frac{1}{25} \implies n^2 = 100 \implies n = 10
Expanding (n−2)(n+2)=n2−4(n-2)(n+2) = n^2 - 4 allows simplifying the algebraic ratio.
3
Evaluate polygon properties for n=10n = 10.
Diagonals: 10(10−3)2=35\frac{10(10-3)}{2} = 35; Interior angle: 8⋅180∘10=144∘\frac{8 \cdot 180^\circ}{10} = 144^\circ; Exterior angle: 360∘10=36∘\frac{360^\circ}{10} = 36^\circ; Interior angle sum: 8⋅180∘=1440∘8 \cdot 180^\circ = 1{}440^\circ.
Apply standard formulas for diagonal count, exterior angle measure, and interior angle sum for a regular decagon.

Key Concept

Interior and exterior angle formulas of regular polygons and diagonal counting formulas
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Topics

Quadrilaterals and Polygons

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