Introduction to Solving Right Triangles
Understanding the properties of a right triangle and its components, such as the legs and the hypotenuse, is fundamental to solving various geometry and trigonometry problems. A right triangle consists of two perpendicular sides (legs) and one diagonal side (hypotenuse) that is the longest side and opposite the right angle.
Given: One Leg of a Triangle Measures 3 cm, the Other Leg is Twice the Measure of the First
This problem involves finding the length of the hypotenuse of a right triangle where one leg is 3 cm and the other is twice its measure, which is 6 cm. This approach is a direct application of the Pythagorean theorem and can also be verified using trigonometric ratios.
Solution Using Pythagorean Theorem
Let's denote the legs of the triangle as a and b, and the hypotenuse as c. Given a 3 cm and b 6 cm, we can find c using the Pythagorean theorem:
[c^2 a^2 b^2] [c^2 3^2 6^2] [c^2 9 36] [c^2 45] [c sqrt{45}] [c 3sqrt{5}] cm
Solution Using Trigonometric Ratios
In addition to the Pythagorean theorem, trigonometric ratios can be used to solve for the hypotenuse in a right triangle. Here, we can use the sine function:
Let A 26.56° and B 63.44°, where C 90°. Using the sine function for the corresponding sides:
[frac{6}{sin(63.44°)} frac{3}{sin(26.56°)} frac{3sqrt{5}}{sin(90°)}] Thus, we can confirm:
[text{Hypotenuse} frac{6}{sin(63.44°)}] [ frac{6}{0.899}] [≈ 6.708] cm
Both methods yield the same result, confirming that the hypotenuse is 3√5 cm, or approximately 6.708 cm.
Conclusion and Additional Applications
The solution provided demonstrates the importance of understanding both the Pythagorean theorem and trigonometric ratios in solving for the hypotenuse of a right triangle. These principles can be applied in various real-life situations, such as construction, architecture, and navigation.
For further exploration, one can delve into more complex right triangle problems involving multiple angles or even non-right triangles. Advanced topics such as the law of sines and the law of cosines can also be explored for these scenarios.