FIND THE PERIMETER OF THE TRIANGLE WHOSE VERTICES ARE: A(3,0), B(5,2), C(7,6)

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!

Calculate the perimeter of a triangle with vertices at A(3,0), B(5,2), and C(7,6) using the distance formula. This problem covers essential geometry concepts such as the distance formula and perimeter calculation. Suitable for students in Grades 8-10.

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