45-45-90 Triangle: Find The Leg Length

Aug 6, 2025 by Pedro Alvarez 39 views

Decoding the Mystery of the 45-45-90 Triangle: Finding the Leg Length

Hey there, math enthusiasts! Today, we're diving into the fascinating world of special right triangles, specifically the 45-45-90 triangle. These triangles are super cool because their sides have a consistent ratio, making them a breeze to work with. We've got a fun problem on our hands: The hypotenuse of a 45-45-90 triangle measures $22 \sqrt{2}$ units. What is the length of one leg of the triangle? Let's break it down together!

Understanding the 45-45-90 Triangle

Before we jump into solving the problem, let's quickly recap what makes a 45-45-90 triangle so special. This special right triangle gets its name from its angles: 45 degrees, 45 degrees, and 90 degrees. The two 45-degree angles mean that this triangle is also an isosceles triangle, meaning it has two sides of equal length. These equal sides are the legs of the right triangle, and the side opposite the right angle (the longest side) is the hypotenuse.

The magic of the 45-45-90 triangle lies in its side ratios. If we let the length of each leg be 'x', then the length of the hypotenuse is always $x\sqrt{2}$ . This ratio is derived from the Pythagorean theorem ( $a^2 + b^2 = c^2$ ), where 'a' and 'b' are the legs and 'c' is the hypotenuse. In our case, $x^2 + x^2 = c^2$ , which simplifies to $2x^2 = c^2$ . Taking the square root of both sides gives us $c = x\sqrt{2}$ .

So, remember this key relationship: leg : leg : hypotenuse = x : x : $x\sqrt{2}$ . This ratio is our secret weapon for solving problems involving 45-45-90 triangles.

Visualizing the Triangle

It's always helpful to visualize what we're working with. Imagine a right triangle that looks like half a square, cut diagonally. The two legs are the sides of the square, and the hypotenuse is the diagonal. The angles formed by the diagonal are both 45 degrees. The visual representation of this triangle can really help solidify the understanding of the relationships between its sides and angles.

The Importance of the Ratio

The 45-45-90 triangle ratio isn't just a mathematical curiosity; it's a powerful tool. This ratio allows us to quickly find the missing side lengths of a 45-45-90 triangle if we know just one side length. For example, if we know the length of a leg, we can easily find the length of the hypotenuse by multiplying it by $\sqrt{2}$ . Conversely, if we know the length of the hypotenuse, we can find the length of a leg by dividing it by $\sqrt{2}$ .

This is incredibly useful in various fields, including trigonometry, geometry, engineering, and even construction. Understanding these special right triangles can significantly simplify problem-solving and save time.

Solving the Problem: Finding the Leg Length

Okay, now that we've refreshed our understanding of the 45-45-90 triangle, let's tackle the problem at hand. We know the hypotenuse measures $22\sqrt{2}$ units, and we need to find the length of one leg. Remember our ratio: leg : leg : hypotenuse = x : x : $x\sqrt{2}$ .

In this case, we know the value of the hypotenuse ( $x\sqrt{2}$ ), which is $22\sqrt{2}$ . So, we can set up the equation:

$x\sqrt{2} = 22\sqrt{2}$

To solve for 'x' (the length of one leg), we simply divide both sides of the equation by $\sqrt{2}$ :

$x = \frac{22\sqrt{2}}{\sqrt{2}}$

The $\sqrt{2}$ terms cancel out, leaving us with:

$x = 22$

Therefore, the length of one leg of the triangle is 22 units.

Step-by-Step Breakdown

Let's recap the steps we took to solve this problem:

Identify the triangle: We recognized that we were dealing with a 45-45-90 triangle.
Recall the ratio: We remembered the side ratio for 45-45-90 triangles: x : x : $x\sqrt{2}$ .
Set up the equation: We equated the given hypotenuse length ( $22\sqrt{2}$ ) to the hypotenuse part of the ratio ( $x\sqrt{2}$ ).
Solve for x: We solved the equation $x\sqrt{2} = 22\sqrt{2}$ for 'x'.
State the answer: We concluded that the length of one leg is 22 units.

Why This Works

The beauty of this solution lies in the consistent ratio within 45-45-90 triangles. Because the angles are fixed, the ratio between the sides remains constant, regardless of the size of the triangle. This allows us to use simple algebra to solve for unknown side lengths, making these triangles incredibly predictable and easy to work with.

Why Option C is the Correct Answer

Looking at the answer choices provided:

A. 11 units
B. $11\sqrt{2}$ units
C. 22 units
D. $22\sqrt{2}$ units

We can clearly see that option C, 22 units, matches our calculated leg length. So, the correct answer is C.

Avoiding Common Mistakes

A common mistake students make is confusing the leg and hypotenuse or misremembering the ratio. It's crucial to remember that the hypotenuse is always the longest side and is opposite the 90-degree angle. Also, double-check the ratio to ensure you're applying it correctly. A simple sketch of the triangle can often help avoid these errors.

The Broader Significance of Special Right Triangles

Understanding special right triangles like the 45-45-90 triangle is more than just solving textbook problems. These triangles pop up everywhere in the real world, from architecture and engineering to computer graphics and even art. They form the basis for many geometric designs and calculations.

Real-World Applications

Architecture: Architects use the properties of 45-45-90 triangles when designing buildings, especially structures with right angles and diagonal supports. These triangles help ensure stability and symmetry.
Engineering: Engineers rely on these triangles for calculations related to bridges, ramps, and other structures where angles and side lengths need precise determination.
Navigation: Navigators use triangles to calculate distances and directions, often employing the principles of special right triangles.
Computer Graphics: In computer graphics, 45-45-90 triangles are used in transformations and projections, allowing for the creation of realistic 3D images.

Beyond the Textbook

The knowledge of special right triangles also lays the groundwork for more advanced concepts in trigonometry and geometry. Understanding the relationships between angles and sides in these triangles is crucial for grasping trigonometric functions like sine, cosine, and tangent.

Practice Makes Perfect

Like any math concept, mastering 45-45-90 triangles requires practice. Try solving various problems where you're given different side lengths and asked to find the missing ones. You can also try drawing your own 45-45-90 triangles and measuring the sides to verify the ratio. Consistent practice will solidify your understanding and make solving these problems second nature.

Resources for Further Learning

If you're eager to learn more about special right triangles and geometry, there are tons of resources available online and in textbooks. Websites like Khan Academy and YouTube channels dedicated to math education offer excellent explanations and practice problems. Exploring these resources can deepen your knowledge and enhance your problem-solving skills.

Conclusion: Mastering the 45-45-90 Triangle

So, there you have it! We've successfully decoded the mystery of the 45-45-90 triangle and found the length of one leg to be 22 units. Remember the key ratio, practice regularly, and you'll be a 45-45-90 triangle master in no time! Understanding these special right triangles not only helps in math class but also provides a valuable foundation for various real-world applications. Keep exploring, keep learning, and keep those math muscles flexing!