We are tasked with finding the length of the diameter of a circle in which triangle $\triangle ABC$ is inscribed, given that:

• $AB = AC = 17$ (isosceles triangle),
• $BC = 16$.

Step 1: Use the circumradius formula for a triangle.

For a triangle inscribed in a circle, the circumradius $R$ is given by: $R = \frac{abc}{4K},$ where $a, b, c$ are the side lengths of the triangle, and $K$ is its area.

The diameter $D$ of the circle is related to $R$ by: $D = 2R.$

Step 2: Use the semiperimeter to find the area $K$.

The semiperimeter $s$ is: $s = \frac{a + b + c}{2} = \frac{17 + 17 + 16}{2} = 25.$

The area $K$ of the triangle is computed using Heron's formula: $K = \sqrt{s(s-a)(s-b)(s-c)}.$ Substitute $a = 17, b = 17, c = 16,$ and $s = 25$: $K = \sqrt{25(25-17)(25-17)(25-16)} = \sqrt{25(8)(8)(9)} = \sqrt{14400} = 120.$

Step 3: Compute the circumradius $R$.

Substitute $a = 17, b = 17, c = 16,$ and $K = 120$ into the circumradius formula: $R = \frac{abc}{4K} = \frac{17 \cdot 17 \cdot 16}{4 \cdot 120} = \frac{4624}{480} = 9.6333 \text{ (approx.)}.$

Step 4: Compute the diameter $D$.

The diameter is: $D = 2R = 2 \cdot 9.6333 = 19.2667.$

Final Answer:

The length of the diameter is approximately: $\boxed{19.27}.$

Would you like a detailed derivation or explanation of any specific step?

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Relative Questions:

1. What is the formula for the circumradius of a triangle in terms of its angles?
2. How does the isosceles property simplify circumcircle calculations?
3. What is the relationship between circumradius and inradius in a triangle?
4. Can the result be verified using a coordinate geometry approach?
5. What happens to the circumradius if $BC$ is doubled?

Tip:

When working with triangles inscribed in circles, ensure that you correctly compute the area using Heron’s formula, as it directly impacts the circumradius calculation.