To solve for the exact value of x using special right triangles, we can leverage the properties of the two primary types of special right triangles: the 45-45-90 triangle and the 30-60-90 triangle.
1. 45-45-90 Triangle
A 45-45-90 triangle has angles of 45°, 45°, and 90°. The sides opposite the 45° angles are equal, and if we denote the length of each leg as a, the hypotenuse will be a√2.
Example: If one leg is known, say a = 5, we can easily find x (hypotenuse):
- Hypotenuse (x) = 5√2
This gives us an exact value of the hypotenuse as 5√2.
2. 30-60-90 Triangle
A 30-60-90 triangle has angles of 30°, 60°, and 90°. The side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is (√3/2) times the hypotenuse.
Example: If we know the length of the side opposite the 30° angle, say a = 4, we can calculate:
- Hypotenuse (x) = 2a = 2 * 4 = 8
- Side opposite the 60° angle = 4√3
Conclusion
By applying these properties, we can solve for x in most problems involving special right triangles. The key is to remember the ratios and relationships that define these triangles. Practice with different values to become comfortable using these methods whenever you encounter right triangles in mathematics!