Triangle ABC can be broken down into two non-congruent right triangles that share coordinates (0,0) and (12,0). If all the sides of the two right triangles are integers, what is the minimum semi-perimeter of ABC

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.

Triangle ABC can be broken down into two non-congruent right triangles that share coordinates (0,0) and (12,0). If all the sides of the two right triangles are integers, what is the minimum semi-perimeter of ABC?

Explore how to calculate the minimum semi-perimeter of triangle ABC, which is formed by two non-congruent right triangles sharing coordinates (0,0) and (12,0), using integer side lengths and Pythagorean triples. This detailed solution covers key geometric concepts and formulas. Suitable for Grades 9-12.

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