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Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
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If a regular polygon has 65 diagonals, then the number of sides this polygon has is:
12
14
13
10
13
Use the formula for diagonals D=2n(n−3) and equate to 65, then solve the quadratic equation or test the given options by substituting n.
Number of diagonals of a regular polygon = 65
D=2n(n−3)
Use the formula for diagonals D=2n(n−3) and equate to 65, then solve the quadratic equation or test the given options by substituting n.
Students often forget to divide by 2 or make a sign error when solving the quadratic equation n2−3n−130=0.
Step 1: Understand the Formula
The number of diagonals D in a polygon with n sides is given by the formula D=2n(n−3).
D=2n(n−3)
Step 2: Substitute the Given Value
We are given that the polygon has 65 diagonals (D=65). Substituting this into the formula gives: 2n(n−3)=65
2n(n−3)=65
Step 3: Solve the Quadratic Equation
Multiply both sides by 2 to clear the fraction: n(n−3)=130 Expanding the left side: n2−3n−130=0
n2−3n−130=0
Step 4: Factorize and Find n
Factor the quadratic equation by finding two numbers that multiply to −130 and add up to −3: (n−13)(n+10)=0 Since the number of sides n must be positive, we take n=13.
n=13
C is correct because substituting n=13 into the diagonal formula yields 213(13−3)=213×10=65, which matches the given number of diagonals.
This formula comes from choosing 2 vertices out of n (nC2=2n(n−1)) and subtracting the n sides of the polygon.
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