Dylan's Metal Triangle: A Simple Geometry Problem Solved

In this article, we explore a fascinating problem involving a piece of metal that Dylan has. Dylan has a square piece of metal that measures 10 inches on each side. He decides to cut the metal along the diagonal, resulting in two right triangles. The question we aim to answer is: what is the length of the hypotenuse of each right triangle to the nearest tenth of an inch? We will use the Pythagorean theorem to solve this problem, making it accessible for students and enthusiasts alike.

Understanding the Geometry

The process begins with analyzing the initial piece of metal. Since Dylan's piece is a square, each side measures 10 inches. When Dylan cuts the square along the diagonal, he creates two identical right triangles. Each right triangle has a base and height of 10 inches, and the diagonal (the hypotenuse of each right triangle) is the line that Dylan draws from one corner of the square to the opposite corner.

Applying the Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that states that for a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it can be represented as:

c^2 a^2 b^2

where c is the length of the hypotenuse, a and b are the lengths of the other two sides.

Solving the Problem Step-by-Step

Let's solve the problem step-by-step using the Pythagorean theorem:

Identify the sides of the triangle: The base and the height of each right triangle are both 10 inches. Let a and b represent the two legs of the right triangle, so a 10 and b 10. Apply the Pythagorean theorem: c^2 10^2 10^2. Calculate the squares: c^2 100 100. Add the squares: c^2 200. Solve for c: c sqrt(200). Calculate the square root of 200: c ≈ 14.14213562. Rounding to the nearest tenth: c ≈ 14.1 inches.

Conclusion

By cutting Dylan's square metal piece along its diagonal, two right triangles are formed, each with a hypotenuse of approximately 14.1 inches to the nearest tenth. This problem not only tests our understanding of the Pythagorean theorem but also reinforces the application of geometric principles in everyday situations.

Frequently Asked Questions (FAQ)

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical law that describes the relationship between the three sides of a right-angled triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is named after the ancient Greek philosopher and mathematician Pythagoras.

Q: How can I apply the Pythagorean theorem in real life?

A: The Pythagorean theorem has numerous real-life applications, such as in construction, navigation, and even in sports. For example, in construction, it can be used to ensure that corners are square, in navigation, it is used to calculate distances, and in sports, it is used to analyze positioning and movement. Understanding and applying it can help solve various practical problems.

Q: Are there other methods to find the length of the diagonal of a square?

A: Yes, there are alternative methods to find the length of the diagonal of a square. One simple method involves using the square root of the sum of the squares of the sides, as we did with the Pythagorean theorem. Another method involves knowing that the diagonal of a square is always equal to the side length multiplied by the square root of two (sqrt(2)). Thus, for a square with side length 10, the diagonal would be 10 * sqrt(2), which is approximately 14.14 inches, again rounding to the nearest tenth gives 14.1 inches.