Let's solve the problem. We are tasked with finding the exact lengths of each side of the triangle $\triangle ABC$, given its vertices:

• $A(0, 0)$,
• $B(3, 8)$,
• $C(6, 0)$.

Afterward, we will verify that $\triangle ABC$ is isosceles by comparing the side lengths.

Step 1: Calculate each side length using the distance formula

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

Length of $AB$:

$AB = \sqrt{(3 - 0)^2 + (8 - 0)^2} = \sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} \, \text{mm}.$

Length of $BC$:

$BC = \sqrt{(6 - 3)^2 + (0 - 8)^2} = \sqrt{(6 - 3)^2 + (-8)^2} = \sqrt{3^2 + 8^2} = \sqrt{73} \, \text{mm}.$

Length of $AC$:

$AC = \sqrt{(6 - 0)^2 + (0 - 0)^2} = \sqrt{6^2} = 6 \, \text{mm}.$

Step 2: Verify if $\triangle ABC$ is isosceles

An isosceles triangle has at least two equal side lengths. From the calculations:

• $AB = \sqrt{73}$,
• $BC = \sqrt{73}$,
• $AC = 6$.

Since $AB = BC$, $\triangle ABC$ is isosceles.

Final Answer:

• $AB = \sqrt{73} \, \text{mm}$,
• $BC = \sqrt{73} \, \text{mm}$,
• $AC = 6 \, \text{mm}$.

$\text{Because } AB = BC, \triangle ABC \text{ is isosceles}.$

Would you like further explanation on the distance formula or the verification process?

Here are five related questions:

1. What is the general formula for verifying an equilateral triangle?
2. How does one calculate the perimeter of $\triangle ABC$?
3. Can you calculate the area of $\triangle ABC$?
4. How would you determine if $\triangle ABC$ is a right triangle?
5. What if $A, B, C$ were given as 3D points? How would the calculations change?

Tip: Always check for at least two equal sides to confirm if a triangle is isosceles.