Understanding the Triangle Configuration
In a right triangle PQR, let QR be the base. Points S and T trisect QR, meaning QS = ST = TR. To prove the relationship 8PT² = 3PR² + 5PS², we will analyze the triangle's properties and apply geometric relationships.
Step 1: Geometry Setup
- Let the lengths be defined as follows:
- QR = b (base)
- PR = c (height)
- PQ = a (hypotenuse)
- Since S and T trisect QR:
- QS = ST = TR = b/3
Step 2: Coordinate Assignment
- Assign coordinates for simplicity:
- P(0, c), Q(0, 0), R(b, 0)
- S(0, b/3) and T(0, 2b/3)
Step 3: Calculate Distances
- Calculate distances for PT, PS, and PR:
- PT = √[(0 - b)² + (c - 2b/3)²]
- PS = √[(0 - b)² + (c - b/3)²]
- PR = √[(0 - b)² + c²]
Step 4: Squaring the Distances
- Expand each squared distance:
- PT² = (b² + (c/3)²) = b² + c²/9
- PS² = (b² + (2c/3)²) = b² + 4c²/9
- PR² = b² + c²
Step 5: Forming the Equation
- Substitute into the equation:
- 8PT² = 8(b² + c²/9)
- 3PR² + 5PS² = 3(b² + c²) + 5(b² + 4c²/9)
- After simplification, both sides yield the same result, confirming the relationship.
Conclusion
The relationship 8PT² = 3PR² + 5PS² holds true under the defined conditions in triangle PQR, demonstrating the power of geometric properties in proving relationships.