45-45-90 Triangles
45-45-90 Triangles is a Grade 8 geometry topic in Saxon Math Course 3, Chapter 7, where students learn the special right triangle with two 45-degree angles whose sides follow the ratio 1 to 1 to sqrt(2). This means the two legs are equal and the hypotenuse is leg times sqrt(2), allowing missing sides to be found instantly without trigonometry.
Key Concepts
Property A 45 45 90 triangle is an isosceles right triangle (half a square) whose side lengths are in the ratio $1:1:\sqrt{2}$.
Examples If the legs of a 45 45 90 triangle are both 1 unit, the hypotenuse is $1 \cdot \sqrt{2} = \sqrt{2}$ units. A baseball diamond is a square with 90 foot sides, so the throw from home to second is the hypotenuse: $90\sqrt{2}$ feet. If one leg of a 45 45 90 triangle measures 8 cm, the other leg is also 8 cm and the hypotenuse is $8\sqrt{2}$ cm.
Explanation Think of this as a perfect square sliced diagonally in half! Because it's an isosceles right triangle, its two legs are always equal. This special relationship means if you know the length of just one leg, you automatically know the other. The hypotenuse is simply the leg length multiplied by the square root of 2.
Common Questions
What are the side ratios of a 45-45-90 triangle?
In a 45-45-90 triangle the two legs are equal and the hypotenuse equals one leg multiplied by the square root of 2. The ratio is 1 : 1 : sqrt(2).
How do you find the hypotenuse of a 45-45-90 triangle?
Multiply the length of either leg by the square root of 2. For example, if each leg is 5, the hypotenuse is 5 times sqrt(2) which is approximately 7.07.
How do you find a leg of a 45-45-90 triangle if you know the hypotenuse?
Divide the hypotenuse by the square root of 2, or equivalently multiply by sqrt(2)/2. This gives the length of each equal leg.
Why is the 45-45-90 triangle called an isosceles right triangle?
It is called isosceles because two sides (the legs) are equal in length, and right because one angle is 90 degrees.
Where are 45-45-90 triangles taught in Grade 8?
They are covered in Saxon Math Course 3, Chapter 7: Algebra, as part of Grade 8 geometry and special right triangles.