To find the largest circle that can fit inside the given isosceles triangle, we need to determine the radius of its incircle. The incircle of a triangle is the largest circle that can fit within the triangle, touching all three sides. Here's how to solve the problem step by step:

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Step 1: Verify the triangle's dimensions

The given sides are $a = 8$, $b = c = 12$. Since $b = c$, this is an isosceles triangle.

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Step 2: Calculate the semi-perimeter

The semi-perimeter ($s$) of the triangle is: $s = \frac{a + b + c}{2} = \frac{8 + 12 + 12}{2} = 16 \, \text{inches.}$

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Step 3: Calculate the area of the triangle

For a triangle with sides $a, b, c$, Heron's formula is used to find the area ($A$): $A = \sqrt{s(s-a)(s-b)(s-c)}.$ Substitute the values: $A = \sqrt{16(16-8)(16-12)(16-12)} = \sqrt{16(8)(4)(4)} = \sqrt{2048} = 32 \, \text{square inches.}$

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Step 4: Use the area to find the radius of the incircle

The radius of the incircle ($r$) is related to the area ($A$) and semi-perimeter ($s$) by: $r = \frac{A}{s}.$ Substitute $A = 32$ and $s = 16$: $r = \frac{32}{16} = 2 \, \text{inches.}$

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Final Answer:

The largest circle that can be cut from the isosceles triangle has a radius of 2 inches.

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Would you like more details on any step or additional examples?
Here are some related questions to expand your understanding:

1. What is the difference between the incircle and circumcircle of a triangle?
2. How would the solution change if the triangle were scalene instead of isosceles?
3. What is Heron's formula, and how is it derived?
4. Can you determine the center of the incircle geometrically?
5. How does the inradius depend on the triangle's proportions?

Tip: Always double-check if a triangle's sides satisfy the triangle inequality before proceeding with calculations.