The 45-45-90 triangle is an isosceles right triangle that is encountered very often in both math and in the world outside of school. It is therefore very useful to get familiar with the properties of this shape. Use the applet below to explore these properties.
a. What relationship do you notice between the leg lengths of the 45-45-90 triangle? b. Use the Pythagorean Theorem to find the hypotenuse of the triangle. How does the hypotenuse compare to the leg lengths of the 45-45-90 triangle? c. Create a ratio comparing one leg length to the other in a 45-45-90 degree triangle (leg/leg, or leg length over leg length). What is the value of this ratio (divide to convert to a decimal)? Does this value change if the size of the triangle increases or decreases? d. Create a ratio comparing either leg length to the hypotenuse length in a 45-45-90 degree triangle (leg/hyp, or leg length over hypotenuse). What is the value of this ratio (divide to convert to a decimal)? Does this value change if the size of the triangle increases or decreases? e. How can you use the leg/hyp ratio to find the length of the hypotenuse if you know the length of the leg (for example, if the leg has a length of 0.75 units)? f. How can you use the leg/hyp ratio to find the length of the leg if you know the length of the hypotenuse (for example, if the hypotenuse has a length of 2.121 units)?