Question: answer this question using this photos i send
1. The number of diagonals of a regular polygon is 65. Find the perimeter of the polygon if its apothem measures 15 inches,
2.The measure of an interior angle of a regular octagon is 135\u00b0. Find the length of the apothem it one side of the polygon measures 6 meters.
3.Find the number of sides of each of the two polygons if the total number of sides of the polygons is 15, and the sum of the number of diagonals of the polygons is 36.
4. A regular polygon has 90 diagonals. Determine the number of sides of the polygon and name it.
1. **Problem:** The number of diagonals of a regular polygon is $65$. Find the perimeter if the apothem is $15$ inches.
2. **Formula for diagonals:** The number of diagonals $D$ of a polygon with $n$ sides is given by:
$$D = \frac{n(n-3)}{2}$$
3. **Find $n$:** Set $D=65$:
$$65 = \frac{n(n-3)}{2} \implies 130 = n(n-3) \implies n^2 - 3n - 130 = 0$$
4. **Solve quadratic:**
$$n = \frac{3 \pm \sqrt{9 + 520}}{2} = \frac{3 \pm \sqrt{529}}{2} = \frac{3 \pm 23}{2}$$
5. **Possible $n$ values:**
$$n = \frac{3 + 23}{2} = 13 \quad \text{or} \quad n = \frac{3 - 23}{2} = -10$$
6. Since $n$ must be positive, $n=13$ sides.
7. **Perimeter $P$ formula:**
$$P = n \times s$$
8. **Find side length $s$ using apothem $a=15$:**
The apothem $a$ relates to side length $s$ by:
$$a = \frac{s}{2 \tan(\pi/n)}$$
9. **Solve for $s$:**
$$s = 2a \tan\left(\frac{\pi}{n}\right) = 2 \times 15 \times \tan\left(\frac{\pi}{13}\right)$$
10. Calculate $\tan(\pi/13)$:
$$\pi/13 \approx 0.2417$$
$$\tan(0.2417) \approx 0.246$$
11. Calculate $s$:
$$s = 30 \times 0.246 = 7.38 \text{ inches}$$
12. Calculate perimeter:
$$P = 13 \times 7.38 = 95.94 \text{ inches}$$
**Final answer:** The perimeter is approximately $95.94$ inches.