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The problem involves proving that the triangle \( \triangle ABC \), with vertices \( A(0, 0) \), \( B(3, 8) \), and \( C(6, 0) \), is isosceles and finding the exact lengths of its sides.

### Step 1: Find the lengths of the sides using the distance formula.

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
\]

#### 1. Length of \( AB \):
Using \( A(0, 0) \) and \( B(3, 8) \):
\[
AB = \sqrt{(3 - 0)^2 + (8 - 0)^2} = \sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73}.
\]

#### 2. Length of \( BC \):
Using \( B(3, 8) \) and \( C(6, 0) \):
\[
BC = \sqrt{(6 - 3)^2 + (0 - 8)^2} = \sqrt{3^2 + (-8)^2} = \sqrt{9 + 64} = \sqrt{73}.
\]

#### 3. Length of \( AC \):
Using \( A(0, 0) \) and \( C(6, 0) \):
\[
AC = \sqrt{(6 - 0)^2 + (0 - 0)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6.
\]

### Step 2: Prove \( \triangle ABC \) is isosceles.
An isosceles triangle has at least two equal sides. From the above calculations:
- \( AB = \sqrt{73} \),
- \( BC = \sqrt{73} \),
- \( AC = 6 \).

Since \( AB = BC \), \( \triangle ABC \) is isosceles.

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### Final Answer:
1. The exact lengths of the sides are:
   - \( AB = \sqrt{73} \),
   - \( BC = \sqrt{73} \),
   - \( AC = 6 \).
2. \( \triangle ABC \) is isosceles because \( AB = BC \).

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Would you like a deeper explanation of the distance formula, or assistance visualizing this triangle on a coordinate plane?

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### Related Questions:
1. What is the perimeter of \( \triangle ABC \) in millimeters?
2. How can you calculate the area of \( \triangle ABC \) using its coordinates?
3. Can this isosceles triangle ever be classified as equilateral? Why or why not?
4. What are the slopes of the sides \( AB \), \( BC \), and \( AC \)?
5. How does the coordinate system influence the physical dimensions of the triangle?

### Tip:
When proving a triangle is isosceles, always compare the exact lengths of its sides carefully using the distance formula!

A manufacturer cuts a piece of metal for a microscope. The resulting piece of metal can be represented in a coordinate plane by a triangle with vertices A(0, 0), B(3, 8), and C(6, 0). One unit in the coordinate plane represents one millimeter. Prove that △ABC is isosceles. Find the exact length of each side.

Learn how to prove that triangle ABC, with vertices A(0, 0), B(3, 8), and C(6, 0), is isosceles. Find the exact lengths of the sides using the distance formula and verify the triangle's properties. Suitable for students in Grades 8-10.

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