The number of diagonals, d, in a polygon of n sides is given by the formula d=n^2-3n/2. If a polygon has 90 diagonals, how many sides has it?
a. 8
b. 10
c. 12
d. 15
e. 20
please answer and explain
just plug in d=90 and solve
(n^2-3n)/2 - 90 = 0
quadratics are easy, right?
I don't think it is easy, can you explain how to do this problem?
Well, first let me just say that polygons are quite the shapes. They've got sides, angles, and even diagonals! Who knew? Now, to find the number of sides in a polygon with 90 diagonals, we can use the formula d = n^2 - 3n/2.
Set d to 90 and let's solve this mystery. We get:
90 = n^2 - 3n/2
Now, we don't want to leave this equation feeling like a square peg in a round hole, so let's multiply everything by 2 to get rid of those pesky fractions:
180 = 2n^2 - 3n
Rearranging things a bit, we get:
2n^2 - 3n - 180 = 0
Oh dear, this equation is giving me a bit of a headache. I think it needs some clowning around to make it more fun. Let's use the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a is 2, b is -3, and c is -180. Plugging those values in, we get:
n = (3 ± √((-3)^2 - 4(2)(-180))) / (2(2))
Hey, look at that! The clown magic is working and the square root part simplifies nicely:
n = (3 ± √(9 + 1440)) / 4
n = (3 ± √1449) / 4
Now, if we keep clowning around with the math, we find two possible solutions for n:
n = (3 + √1449) / 4 or n = (3 - √1449) / 4
Taking a closer look, we find that n has to be a positive whole number. Unfortunately, (3 - √1449) / 4 is negative, so that's not a winner. But fear not! (3 + √1449) / 4 does give us a positive result.
After some more clownish calculations, we find that n ≈ 15. So, the polygon with 90 diagonals has 15 sides. And the answer is... (drumroll, please)... d. 15!
Happy clowning, my friend!
To find out how many sides a polygon has given the number of diagonals, we can use the formula d = n^2 - 3n/2, where d represents the number of diagonals and n represents the number of sides.
In this case, we are given that the polygon has 90 diagonals. So we can set up the equation as follows:
90 = n^2 - 3n/2
To solve this equation, we can multiply both sides by 2 to get rid of the fraction:
180 = 2n^2 - 3n
Rearranging the equation, we have a quadratic equation:
2n^2 - 3n - 180 = 0
This equation can be factored or solved using the quadratic formula. Let's factorize it:
(2n + 15)(n - 12) = 0
Setting each factor equal to zero, we have two possible solutions:
2n + 15 = 0 or n - 12 = 0
Solving for n in each case:
2n = -15 or n = 12
n = -15/2 is not a valid solution since we cannot have a negative number of sides for a polygon.
Therefore, the polygon has 12 sides. Thus, the correct answer is option c. 12.
You have a formula for finding d, given n. You are given d, so you need to find n.
(n^2-3n)/2 = 90
n^2-3n = 180
n^2 - 3n - 180 = 0
(n-15)(n+12) = 0
n = 15 or -12
-12 is not a reasonable answer, so n=15 is the solution.