CAT 2023 Slot 3 QA Question 19

Type-in-the-answer (no negative marking) · Geometry · Polygons · Try it, then check the answer and solution below.

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

TITA Answer:

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Official Convenor Answer Key: 54

Answer and solution

Answer: 54

📌 Core Concept

In a regular polygon, the sum of an interior angle and its corresponding exterior angle is

180^\circ

. The number of sides

n

of a regular polygon can be found using the formula:

n = \frac{360^\circ}{\text{measure of each exterior angle}}

The number of diagonals in a polygon with

n

sides is given by:

\text{Number of diagonals} = \frac{n(n - 3)}{2}

🔢 Step-by-Step Solution

Let the exterior angle be x$ degrees.

Given that the interior angle exceeds the exterior angle by 120 degrees:

\text{Interior angle} = x + 120^\circ

Since interior and exterior angles are supplementary:

x + (x +

120^\circ

) =

180^\circ$

Solve for x$:

2x + 120^\circ = 180^\circ \\ 2x = 60^\circ \\ x = 30^\circ

Calculate the number of sides n$:

n = \frac{360^\circ}{30^\circ} = 12

Find the number of diagonals using the formula:

\text{Number of diagonals} = \frac{12(12 - 3)}{2} = \frac{12 × 9}{2} = \frac{108}{2} = 54

⚡ 30-Second Shortcut

Exterior angle = 30°, so number of sides = 360 / 30 = 12.

Diagonals formula = n(n - 3)/2 = 12×9/2 = 54.

🎯 Final Answer

Correct Answer: 54

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