11. a. How many diagonals are in a 53 -gon? b. Name the polygon that has 1,710 diagonals.

Question

UpStudy AI · Accepted Answer

To find the number of diagonals in a polygon, you can use the formula:

$$
D = \frac{n(n - 3)}{2}
$$

where $$D$$ is the number of diagonals and $$n$$ is the number of sides (or vertices) of the polygon.

### a. For a 53-gon:

1. Substitute $$n = 53$$ into the formula:

$$
D = \frac{53(53 - 3)}{2}
$$

2. Calculate $$53 - 3$$:

$$
53 - 3 = 50
$$

3. Now substitute back into the formula:

$$
D = \frac{53 \times 50}{2}
$$

4. Calculate $$53 \times 50$$:

$$
53 \times 50 = 2650
$$

5. Now divide by 2:

$$
D = \frac{2650}{2} = 1325
$$

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:

$$
\frac{n(n - 3)}{2} = 1710
$$

1. Multiply both sides by 2:

$$
n(n - 3) = 3420
$$

2. Rearranging gives us a quadratic equation:

$$
n^2 - 3n - 3420 = 0
$$

3. We can use the quadratic formula $$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$, where $$a = 1$$, $$b = -3$$, and $$c = -3420$$:

$$
n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-3420)}}{2 \cdot 1}
$$

4. Calculate the discriminant:

$$
(-3)^2 - 4 \cdot 1 \cdot (-3420) = 9 + 13680 = 13689
$$

5. Now take the square root:

$$
\sqrt{13689} = 117
$$

6. Substitute back into the quadratic formula:

$$
n = \frac{3 \pm 117}{2}
$$

This gives us two potential solutions:

$$
n = \frac{120}{2} = 60 \quad \text{and} \quad n = \frac{-114}{2} = -57
$$

Since $$n$$ must be a positive integer, we have:

$$
n = 60
$$

Thus, the polygon that has 1,710 diagonals is a **60-gon**.
a. A 53-gon has 1325 diagonals. b. A polygon with 1,710 diagonals is a 60-gon.

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