Question
- a. How many diagonals are in a 53 -gon?
b. Name the polygon that has 1,710 diagonals.
b. Name the polygon that has 1,710 diagonals.
Ask by Bates West. in the United States
Nov 20,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
a. A 53-gon has 1325 diagonals.
b. A polygon with 1,710 diagonals is a 60-gon.
b. A polygon with 1,710 diagonals is a 60-gon.
Solution
To find the number of diagonals in a polygon, you can use the formula:
where
is the number of diagonals and
is the number of sides (or vertices) of the polygon.
a. For a 53-gon:
- Substitute
into the formula:
- Calculate
:
- Now substitute back into the formula:
- Calculate
:
- Now divide by 2:
So, the number of diagonals in a 53-gon is 1325.
b. To find the polygon with 1,710 diagonals:
We can set up the equation using the diagonal formula:
- Multiply both sides by 2:
- Rearranging gives us a quadratic equation:
- We can use the quadratic formula
, where , , and :
- Calculate the discriminant:
- Now take the square root:
- Substitute back into the quadratic formula:
This gives us two potential solutions:
Since
must be a positive integer, we have:
Thus, the polygon that has 1,710 diagonals is a 60-gon.
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The Deep Dive
To find the number of diagonals in a polygon, you can use the formula: Diagonals = n(n - 3)/2, where n is the number of sides. For a 53-gon, plugging in the values gives you: 53(53 - 3)/2 = 1,275 diagonals.
Now, if you’re looking for a polygon with 1,710 diagonals, you can rearrange the same formula to find n. Solving the equation n(n - 3)/2 = 1,710 leads you to find that the polygon is a 66-gon. It’s always fun to explore the relationships between polygons and their properties!