Properties of a 45°-45°-90° Triangle
Use 45-45-90 triangle properties to find side lengths: legs are equal and the hypotenuse equals leg times sqrt(2), enabling fast calculations without the full Pythagorean Theorem.
Key Concepts
In a $45° 45° 90°$ right triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg multiplied by $\sqrt{2}$. The side ratios follow a $1 1 \sqrt{2}$ pattern.
If a leg is $7$ cm, the hypotenuse is $7\sqrt{2}$ cm. If the hypotenuse is $10\sqrt{2}$ feet, each leg is $\frac{10\sqrt{2}}{\sqrt{2}} = 10$ feet. If a leg measures $9$, the hypotenuse is $9\sqrt{2}$.
Think of this triangle as a perfect square sliced in half diagonally. This is why its two legs are always equal! To find the hypotenuse (the long side), just take the length of a leg and multiply it by $\sqrt{2}$. It’s a super shortcut that saves you from doing the full Pythagorean theorem every single time.
Common Questions
What are the side length ratios of a 45-45-90 triangle?
In a 45-45-90 triangle the two legs are equal in length and the hypotenuse equals one leg multiplied by sqrt(2). If each leg has length s, the hypotenuse is s*sqrt(2). This ratio holds for every isosceles right triangle regardless of size.
How do you find the leg length when given the hypotenuse of a 45-45-90 triangle?
Divide the hypotenuse by sqrt(2), or equivalently multiply it by sqrt(2)/2. For a hypotenuse of 10, legs have length 10/sqrt(2) = 5*sqrt(2) which is approximately 7.07.
Where do 45-45-90 triangles appear in Grade 10 Saxon Algebra 2?
These special right triangles appear when evaluating trigonometric functions at 45 degrees, simplifying radical expressions, and solving geometry problems involving squares cut along a diagonal. Memorizing the ratio eliminates repeated Pythagorean calculations.