The Number Of Diagonals Of A Polygon Of n Sides Is Given By The Formula GMAT Problem Solving

Question: The number of diagonals of a polygon of n sides is given by the formula d=n(n-3)/2. If a polygon has twice as many diagonals as sides, how many sides does it have?

(A) 3
(B) 5
(C) 6
(D) 7
(E) 8

“The number of diagonals of a polygon of n sides is given by the formula''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT quant section examines the analytical and mathematical aptitudes of the candidates. The students must choose the correct option by interpreting the mathematical problems with proper calculations. The students need to have better knowledge of mathematics to solve GMAT Problem Solving questions. The calculative mathematical problems based on the GMAT Quant topic in the problem-solving part can be solved with proper quantitative aptitudes. The candidates can practice questions by answering from the book “GMAT Official Guide 2021”.

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:

Given:

• The number of diagonals of a polygon of n sides is presented by the formula d=n(n-3)/2.
• The polygon has twice as many diagonals as its sides.

Find Out:

• The number of sides the polygon has.

As per the given formula of the number of diagonals of a polygon of n sides, d=n(n-3)/2.

Let’s test the equation by assuming 5 sides of a polygon.
d= 5(5-3)/2
Or, d= 5*2/2
Or, d =5

However, the value of d=5 is irrelevant since the polygon has twice as many diagonals as its sides, i.e d=2n. Here 5 is not a multiple of 2.

If the equation is tested by assuming 7 sides of a polygon,
then d= 7(7-3)/2
Or, d= 7* 4/2
Or, d= 14

Therefore, d=14 is sufficient since 14 is the multiple of 2 and it satisfies the equation d=2n.
Hence, the number of sides the polygon has= 7

Correct Answer: (D)

Approach Solution 2:

The problem statement suggests that:

Given:

• The number of diagonals of a polygon of n sides is presented by the formula d=n(n-3)/2.
• The polygon has twice as many diagonals as its sides.

Find Out:

• The number of sides the polygon has.

The formula of the number of diagonals of a polygon of n sides is d= n(n-3)/2
Since the polygon has n sides and as per the given condition of the question, d=2n
Therefore we can infer that,
2n= n(n-3)/2
Or, 4n= n^2 - 3n
Or, 7n= n^2
Or, n=7

Hence, the number of sides the polygon has= 7

Correct Answer: (D)

Approach Solution 3:

The problem statement indicates that:

Given:

• The number of diagonals of a polygon of n sides is illustrated by the formula d=n(n-3)/2.
• The polygon has twice as many diagonals as its sides.

Find Out:

• The number of sides the polygon has.

The number of diagonals of a polygon of n sides, d=n(n-3)/2.
Given, n(n-3)/2 =2n (since the diagonal is twice the sides of a polygon)

Let’s check the options to answer this question.

(A) 3 i.e n(n-3)/2 = 3*0/2 =0 which is not twice the sides of a polygon. Therefore, it is incorrect.
(B) 5 i.e n(n-3)/2 = 5*2/2 =5 which is not twice the sides of a polygon. Therefore, it is inaccurate.
(C) 6 i.e n(n-3)/2 = 6*3/2 =9 which is not twice the sides of a polygon. Therefore, it is wrong.
(D) 7 i.e n(n-3)/2 = 7*4/2 =14 which is twice the sides of a polygon. Therefore, it is correct.
(E) 8, i.e. so we can eliminate this option.

Correct Answer: (D)

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