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CAT 2023 Slot 3 — QA Question 19

PolygonsEasy

In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagonals of this polygon is

Answer & solution

Answer: 54

Solution

Easy

Interior and exterior angles are supplementary (int+ext=180∘\text{int}+\text{ext}=180^\circ). Use the 120∘120^\circ gap to find both, get the number of sides nn from the interior-angle formula, then count diagonals with n(n−3)2\dfrac{n(n-3)}{2}.

1

Find the exterior angle. Let it be x∘x^\circ; the interior angle is (x+120)∘(x+120)^\circ, and they sum to 180∘180^\circ:

x+(x+120)=180 ⇒ 2x=60 ⇒ x=30∘x+(x+120)=180\ \Rightarrow\ 2x=60\ \Rightarrow\ x=30^\circ

So the interior angle =30+120=150∘=30+120=150^\circ.

2

Find the number of sides from the regular-polygon interior angle (n−2)n⋅180∘\dfrac{(n-2)}{n}\cdot 180^\circ:

(n−2)n⋅180=150 ⇒ n−2n=56⇒ 6n−12=5n ⇒ n=12\begin{aligned} &\frac{(n-2)}{n}\cdot 180=150\ \Rightarrow\ \frac{n-2}{n}=\frac{5}{6}\\ &\Rightarrow\ 6n-12=5n\ \Rightarrow\ n=12 \end{aligned}
3

Count the diagonals of a 1212-gon:

n(n−3)2=12⋅92=54\frac{n(n-3)}{2}=\frac{12\cdot 9}{2}=54
number of diagonals=54\text{number of diagonals}=\mathbf{54}

Exterior angle =30∘=30^\circ instantly gives n=360∘30∘=12n=\dfrac{360^\circ}{30^\circ}=12 — skip the interior-angle formula entirely.

CAT 2023 Slot 3 QA Q19: In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagon — Solution | TheCATExam