Finding The Leg Length Of A 45-45-90 Triangle With Hypotenuse 7√2
Hey guys! Let's dive into a fascinating world of geometry, specifically focusing on a special type of triangle: the 45-45-90 triangle. These triangles are super cool because they have some really neat properties that make solving problems involving them much easier. In this article, we're going to tackle a common question: If a 45-45-90 triangle has a hypotenuse of 7√2 units, what's the length of one of its legs? Get ready to unlock the secrets of these triangles and become a geometry whiz!
Understanding the 45-45-90 Triangle
Before we jump into solving the problem, let's make sure we're all on the same page about what a 45-45-90 triangle actually is. As the name suggests, it's a right triangle (meaning it has one 90-degree angle) where the other two angles are both 45 degrees. This unique angle combination gives it some special characteristics. The most important thing to remember is the relationship between its sides: the legs are congruent (equal in length), and the hypotenuse is √2 times the length of a leg.
Let's break that down a little further. Imagine we have a 45-45-90 triangle. Let's call the length of each leg 'x'. Because the legs are equal, they both have a length of 'x'. Now, the hypotenuse, which is the side opposite the right angle and the longest side of the triangle, has a length of x√2. This relationship is the key to solving many problems involving these triangles.
This relationship stems from the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): a² + b² = c². In our 45-45-90 triangle, a = x, b = x, and c = x√2. If we plug these values into the Pythagorean Theorem, we get:
x² + x² = (x√2)²
Simplifying this equation, we get:
2x² = 2x²
This confirms that the relationship we stated earlier is indeed true and derived from the fundamental Pythagorean Theorem. Understanding this connection is crucial for truly grasping the properties of 45-45-90 triangles and applying them effectively.
So, to recap, in a 45-45-90 triangle:
- The two legs are always equal in length.
- The hypotenuse is always √2 times the length of a leg.
With this knowledge in our toolbox, we're ready to tackle the problem at hand!
Solving for the Leg Length
Okay, guys, now let's get down to business and solve the problem. We know that the hypotenuse of our 45-45-90 triangle measures 7√2 units. Our mission is to find the length of one of the legs. Remember that key relationship we just talked about? The hypotenuse is √2 times the length of a leg. We can use this to our advantage.
Let's represent the length of one leg as 'x', just like we did before. Then, according to our relationship, the hypotenuse is x√2. We're given that the hypotenuse is 7√2, so we can set up an equation:
x√2 = 7√2
Now, we just need to solve for 'x'. To isolate 'x', we can divide both sides of the equation by √2:
(x√2) / √2 = (7√2) / √2
The √2 terms cancel out on both sides, leaving us with:
x = 7
Boom! We've found our answer. The length of one leg of the triangle is 7 units. This result highlights the elegance and simplicity of working with 45-45-90 triangles. By understanding the fundamental relationship between their sides, we can quickly solve for unknown lengths without needing complex calculations.
Let's think about why this works so well. The √2 factor is the linchpin. It directly connects the leg length to the hypotenuse length. When we divide the hypotenuse (7√2) by √2, we effectively
