Determine the Length of the Third Side in a Right-Angled Triangle
In this article, we will explore how to calculate the length of the third side in a right-angled triangle when given the lengths of the other two sides. Specifically, we will consider a triangle ABC where angle B is a right angle (90 degrees) and the sides adjacent to this angle are AB 6 meters and BC 8 meters.
Understanding the Right-Angled Triangle
A right-angled triangle is a triangle that has one angle measuring exactly 90 degrees. The side opposite the right angle is called the hypotenuse, which is the longest side of the triangle.
Applying the Pythagorean Theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
The formula for the Pythagorean theorem is:
[ c^2 a^2 b^2 ]
Step-by-Step Calculation
Given the sides of the triangle, we have:
AB 6 meters (one side) BC 8 meters (the other side) We need to find AC (the hypotenuse)Using the Pythagorean theorem, we can set up the equation as follows:
[ AC^2 AB^2 BC^2 ]
Substituting the given values:
[ AC^2 6^2 8^2 ]
Calculating the squares:
[ AC^2 36 64 ]
[ AC^2 100 ]
Next, we take the square root of both sides:
[ AC sqrt{100} ]
[ AC 10 text{ meters} ]
Conclusion
Therefore, the length of the third side (hypotenuse) AC in the triangle ABC, where angle B is 90 degrees, AB 6 meters, and BC 8 meters, is 10 meters.
By applying the Pythagorean theorem, we can solve various problems related to right-angled triangles. This method is widely used in geometry, trigonometry, and real-world applications like construction and navigation.
Additional Examples and Practice
Understanding the Pythagorean theorem is crucial for solving more complex problems involving right-angled triangles. Here are a few examples and practice problems:
If a 5 meters and b 12 meters, what is the length of c? If a 8 meters and c 17 meters, what is the length of b? If c 15 meters and b 9 meters, what is the length of a?Practice these problems to enhance your understanding and proficiency in using the Pythagorean theorem.