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Geometry is full of fascinating shapes and theorems, and one such important figure is the 45-45-90 triangle. In this blog post, we will explore the theorem behind these triangles, provide a step-by-step calculation guide, offer practical examples, and discuss common mistakes. So, let’s dive into the world of 45-45-90 triangles!

Introduction

A 45-45-90 triangle is a special type of right triangle where two of its angles are equal, measuring 45 degrees each. These triangles have unique properties that make them valuable in solving geometric problems and real-life scenarios.

The Theorem Behind 45-45-90 Triangles

To understand 45-45-90 triangles, it’s important to grasp the theorem that governs their properties. This theorem states that in a 45-45-90 triangle, the lengths of the sides have a specific relationship: the length of the hypotenuse is equal to √2 times the length of each leg.

Step-by-Step Calculation Guide

Now, let’s explore how to calculate the lengths of the sides in a 45-45-90 triangle. We will walk through the identification of these triangles, determining the lengths using the theorem, and solving for the hypotenuse and legs.

Step 1: Identifying a 45-45-90 Triangle

The first step is to confirm that you are indeed working with a 45-45-90 triangle. This can be determined by identifying two equal angles of 45 degrees, which automatically signifies that the third angle is 90 degrees, making it a right triangle.

Step 2: Using The Theorem to Determine Lengths

Once you’ve identified the triangle, refer to the theorem for 45-45-90 triangles, which states: the hypotenuse is √2 times the length of one of the legs (since both legs are of equal length). For instance, if one leg (we’ll call it ‘a’) is 5 units long, the hypotenuse (c) can be calculated as \( c = a \times \sqrt{2} \).

Step 3: Calculating the Hypotenuse

Use the theorem to calculate the hypotenuse if the length of one leg is known. For example, if one leg is 5 units, then the hypotenuse would be 5 × √2 ≈ 7.07 units.

Step 4: Solving for the Legs

If the hypotenuse’s length is given and you’re asked to find the lengths of the legs, work backward using the theorem. Divide the length of the hypotenuse by √2 to find the length of one leg. For instance, if the hypotenuse is 7.07 units, each leg would be 7.07 / √2 ≈ 5 units.

Practical Examples

To solidify our understanding, let’s work through a few practical examples. We will apply the theorem to solve problems involving 45-45-90 triangles and demonstrate how these triangles show up in real-life scenarios.

Example 1: Calculating the Hypotenuse

Imagine you’re creating a square kite, and each side of the square measures 4 feet. To add stability, you decide to place a diagonal support. This diagonal essentially halves the square into two 45-45-90 triangles. To determine the length of this diagonal (which is also the hypotenuse of the triangle), apply the theorem:

For a leg (a) = 4 feet,

Hypotenuse (c) = \( a \times \sqrt{2} = 4 \times \sqrt{2} \approx 5.66 \) feet.

This means you’ll need a diagonal support of approximately 5.66 feet to stabilize the kite.

Example 2: Finding the Length of the Legs

During a homework assignment, you’re provided with a right-angled triangle with an hypotenuse measuring 14 units and are told it’s a 45-45-90 triangle. To find the lengths of the legs, you use the theorem in reverse:

Given hypotenuse (c) = 14 units,

Length of one leg (a) = \( \frac{c}{\sqrt{2}} = \frac{14}{\sqrt{2}} \approx 9.89 \) units.

Thus, each leg of the triangle measures approximately 9.89 units.

These examples illustrate the practical application of the 45-45-90 triangle theorem, whether it’s crafting a kite or solving geometric problems in homework, showcasing the theorem’s utility in both theoretical and real-life contexts.

Common Mistakes and How to Avoid Them

As with any mathematical concept, there are common errors that students may make when calculating 45-45-90 triangles. We’ll discuss these mistakes and provide tips and tricks to help you avoid them, ensuring accurate calculations and a deeper understanding of the concept.

One frequent mistake students make is confusing the roles of the sides in the theorem application, especially under exam pressure. It’s critical to remember that the theorem specifically applies to the hypotenuse being √2 times the length of a leg—not the other way around. This misapplication can lead to incorrect answers and confusion in more complex geometric problems.

Another common error involves miscalculating the √2 factor when solving for the hypotenuse or the legs. Given that √2 is an irrational number, rounding it too early in your calculations can lead to inaccuracies. Always use the √2 factor in its complete form and only round off the final answer to maintain precision.

Lastly, students occasionally assume all right triangles are 45-45-90 triangles, which is not the case. This assumption can lead to incorrect applications of the theorem to scenarios where it doesn’t apply. It’s essential to verify the two 45 degree angles before proceeding with calculations based on this specific triangle’s properties.

Conclusion

Congratulations! You’ve reached the end of our step-by-step guide to calculating 45-45-90 triangles. Now, it’s time to put your knowledge into practice. Keep practicing, solving problems, and exploring geometry, and you’ll continue to strengthen your mathematical skills.

Remember, mastering 45-45-90 triangles is just the beginning of your mathematical journey. As you progress, you’ll encounter many more fascinating concepts and theorems. So, embrace the challenge, stay curious, and enjoy the beauty of mathematics!