Take π and 22/7 all the time and find the arc length and radius separately and explain pls

The perimeter $P$ of a sector is given by the sum of its two radii and its arc length $L$:

$$P = 2r + L$$

We are given $P = 260$ cm.

The sector is $\frac{2}{3}$ of a circle. The circumference of a full circle is $C = 2\pi r$.
Therefore, the arc length $L$ is $\frac{2}{3}$ of the circumference:

$$L = \frac{2}{3} (2\pi r)$$

Substitute $\pi = \frac{22}{7}$:

$$L = \frac{2}{3} \left(2 \times \frac{22}{7} \times r\right)$$ $$L = \frac{2}{3} \left(\frac{44}{7} r\right)$$ $$L = \frac{88}{21} r$$

This is the expression for the arc length in terms of $r$.

Substitute the expression for $L$ into the perimeter formula:

$$260 = 2r + \frac{88}{21} r$$

Factor out $r$:

$$260 = r \left(2 + \frac{88}{21}\right)$$

Find a common denominator for the terms inside the parenthesis:

$$260 = r \left(\frac{2 \times 21}{21} + \frac{88}{21}\right)$$ $$260 = r \left(\frac{42}{21} + \frac{88}{21}\right)$$ $$260 = r \left(\frac{42 + 88}{21}\right)$$ $$260 = r \left(\frac{130}{21}\right)$$

Now, solve for $r$:

$$r = \frac{260 \times 21}{130}$$ $$r = 2 \times 21$$ $$r = 42$$

The radius of the sector is $\boxed{\text{42 cm}}$.

Now that we have the radius $r = 42$ cm, we can calculate the arc length $L$ using the formula from Step 2:

$$L = \frac{88}{21} r$$

Substitute $r = 42$:

$$L = \frac{88}{21} \times 42$$ $$L = 88 \times \frac{42}{21}$$ $$L = 88 \times 2$$ $$L = 176$$

The arc length of the sector is $\boxed{\text{176 cm}}$.