Understanding the Right-Angle Triangle
In a right-angle triangle, we have three vertices: A, B, and C, where angle C is the right angle. The side opposite the right angle, AB, is the hypotenuse.
Midpoint of the Hypotenuse
Let D be the midpoint of hypotenuse AB. We need to prove that D is equidistant from points A and C, and points B and C.
Distance Formula
To show that D is equidistant from A and C, and from B and C, we can use the distance formula.
1. Coordinates of the Points
Assume:
- A (0, 0)
- B (c, 0)
- C (0, b)
Then, the coordinates of D, the midpoint of AB, can be calculated as:
- D = ((0 + c)/2, (0 + 0)/2) = (c/2, 0)
2. Distance from D to A
The distance from D to A is:
- DA = √(((c/2) - 0)² + (0 - 0)²) = c/2
3. Distance from D to B
The distance from D to B is:
- DB = √(((c/2) - c)² + (0 - 0)²) = c/2
4. Distance from D to C
The distance from D to C is:
- DC = √(((c/2) - 0)² + (0 - b)²) = √((c/2)² + b²)
Conclusion
In a right-angle triangle, the midpoint of the hypotenuse is equidistant from the vertices. Thus, DA = DB = DC, proving that D is indeed equidistant from points A and C, and B and C. This property is essential in various geometric applications and helps in understanding triangle properties.