To find the minimum semi-perimeter of triangle $ABC$ that can be broken down into two non-congruent right triangles with integer side lengths, sharing the coordinates $(0,0)$ and $(12,0)$, we proceed as follows:

Step 1: Understanding the Problem

• Given points: $(0,0)$, $(12,0)$.
• Right triangles: We need two right triangles with integer side lengths that share these points as two of their vertices.
• Objective: Minimize the semi-perimeter of triangle $ABC$, which is composed of these two triangles.

Step 2: Setup of the Triangles

Let's define:

• The first triangle $\triangle ABD$ with vertices $(0,0)$, $(12,0)$, and $(a,b)$.
• The second triangle $\triangle BCD$ with vertices $(0,0)$, $(12,0)$, and $(c,d)$.

Here, $(a, b)$ and $(c, d)$ are the other vertices of the right triangles. They must be such that the triangles formed are right triangles with integer side lengths.

Step 3: Finding Suitable Integer Points

To find the right triangles, we can use Pythagorean triples. We want both $(a, b)$ and $(c, d)$ to form right triangles with the base segment $(12, 0)$.

Pythagorean Triple for $(a, b)$

Assuming $(a, b)$ forms a right triangle with the segment of length 12:

• We seek integers $a$ and $b$ such that: $a^2 + b^2 = 12^2 = 144$

Among the integer solutions, $(a, b) = (9, 12)$ satisfies this equation. This corresponds to the first triangle.

Pythagorean Triple for $(c, d)$

Now, we find another set of integers $(c, d)$ such that $c^2 + d^2 = 12^2 = 144$ and the triangle formed is not congruent to the previous one. A suitable pair is $(5, 12)$. This gives us the second triangle.

Step 4: Semi-Perimeter Calculation

• First triangle $\triangle ABD$:

  • Sides: $9, 12, 15$ (where 15 is the hypotenuse from $(a, b) = (9, 12)$).
  • Perimeter: $9 + 12 + 15 = 36$.
  • Semi-perimeter: $\frac{36}{2} = 18$.

• Second triangle $\triangle BCD$:

  • Sides: $5, 12, 13$ (where 13 is the hypotenuse from $(c, d) = (5, 12)$).
  • Perimeter: $5 + 12 + 13 = 30$.
  • Semi-perimeter: $\frac{30}{2} = 15$.

Step 5: Semi-Perimeter of Triangle $ABC$

The semi-perimeter of triangle $ABC$ is the sum of the semi-perimeters of the two right triangles: $\text{Semi-perimeter of } ABC = 18 + 15 = 33$

Conclusion

The minimum semi-perimeter of triangle $ABC$ is 33.

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Would you like a deeper explanation or have any questions about the solution?

Related Questions:

1. What are Pythagorean triples and how are they used in solving geometric problems?
2. How can the semi-perimeter of a triangle be applied in finding other properties such as the area?
3. Can there be another set of integer side lengths resulting in a smaller semi-perimeter?
4. How does the use of coordinates help in solving triangle problems with integer constraints?
5. What methods exist for identifying congruent versus non-congruent triangles using coordinates?

Tip: When solving problems involving integer coordinates and triangles, using known Pythagorean triples can significantly simplify finding right triangles with integer side lengths.