The 45-45-90 triangle theorem states that in an isosceles right triangle, the angles measure 45 degrees, 45 degrees, and 90 degrees, and the lengths of the legs are equal while the length of the hypotenuse is $ ext{leg} \cdot \sqrt{2}$. This theorem is essential for solving problems involving special right triangles and helps establish relationships between their sides and angles.
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In a 45-45-90 triangle, both legs are of equal length, denoted as 'x'. Thus, the hypotenuse will be 'x√2'.
The ratio of the lengths of the sides in a 45-45-90 triangle is always 1:1:√2.
You can use the 45-45-90 triangle theorem to derive lengths quickly when given one leg's length.
This theorem is particularly useful in geometry proofs and real-world applications involving right angles.
The angles of a 45-45-90 triangle add up to 180 degrees, maintaining the fundamental property of triangles.
Review Questions
How can you apply the 45-45-90 triangle theorem to find missing side lengths in geometric problems?
To find missing side lengths using the 45-45-90 triangle theorem, you can start by identifying one leg's length. Since both legs are equal in this type of triangle, if one leg measures 'x', the other will also be 'x', and the hypotenuse can be calculated as 'x√2'. This allows you to quickly solve for unknown sides when working with problems involving isosceles right triangles.
Compare the properties of a 45-45-90 triangle with those of a 30-60-90 triangle in terms of side ratios and applications.
A 45-45-90 triangle has side ratios of 1:1:√2, meaning both legs are equal, while a 30-60-90 triangle has ratios of 1:√3:2. The applications for both triangles are often seen in trigonometry and geometry, but they serve different purposes. The 45-45-90 triangle is typically used when dealing with symmetry and diagonal problems, whereas the 30-60-90 triangle often arises in scenarios involving split angles or specific height calculations.
Evaluate how understanding the properties of special right triangles like the 45-45-90 theorem enhances problem-solving abilities in more complex geometric scenarios.
Understanding special right triangles such as the 45-45-90 theorem allows for efficient problem-solving by providing quick reference points for side lengths and angle relationships. This knowledge aids in tackling more complex geometric scenarios by breaking them down into simpler components. For example, when analyzing shapes that include these triangles or when applying trigonometric functions, having a strong grasp of their properties enables you to approach problems with confidence and streamline calculations effectively.
Related terms
Isosceles Triangle: A triangle with at least two equal sides and two equal angles.
Pythagorean Theorem: A fundamental relation in Euclidean geometry among the three sides of a right triangle, stating that $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
Special Right Triangles: Triangles with specific angle measures (like 30-60-90 and 45-45-90) that allow for easy calculations of side lengths.