To find the perimeter of the triangle with vertices $A(3, 0)$, $B(5, 2)$, and $C(7, 6)$, we first need to calculate the lengths of all three sides using the distance formula.

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 1: Calculate $AB$

Using points $A(3, 0)$ and $B(5, 2)$:

$AB = \sqrt{(5 - 3)^2 + (2 - 0)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

Step 2: Calculate $BC$

Using points $B(5, 2)$ and $C(7, 6)$:

$BC = \sqrt{(7 - 5)^2 + (6 - 2)^2} = \sqrt{(2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$

Step 3: Calculate $AC$

Using points $A(3, 0)$ and $C(7, 6)$:

$AC = \sqrt{(7 - 3)^2 + (6 - 0)^2} = \sqrt{(4)^2 + (6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}$

Step 4: Calculate the Perimeter

The perimeter $P$ of the triangle is the sum of all three sides:

$P = AB + BC + AC = 2\sqrt{2} + 2\sqrt{5} + 2\sqrt{13}$

We can factor out the 2:

$P = 2(\sqrt{2} + \sqrt{5} + \sqrt{13})$

This is the exact perimeter of the triangle. If you prefer an approximate value, you can calculate:

$P \approx 2(1.414 + 2.236 + 3.606) = 2(7.256) \approx 14.512$

Thus, the approximate perimeter of the triangle is $\approx 14.512$ units.

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do you find the area of a triangle with given vertices?
2. What is the formula for calculating the distance between two points?
3. How do you find the slope of a line between two points?
4. What is the difference between exact and approximate values in geometry?
5. How do you classify triangles by their sides and angles?

Tip: Always double-check your calculations, especially when working with square roots and distances!