Rigid Transformations- transformations that don’t change the shape or size of an object; also called an isometry- the preimage and image are congruent.
The three types of rigid transformations are reflections, rotations, and translations.
We practiced applying each transformation with different methods:
- Using appropriate tools/strategies to facilitate the process (tracing, folding, measuring)
- Just “eyeballing it”
- Using properties of the transformation to create accurate information (Perpendicular bisectors for reflections, equal distances and angles for rotations)
We then worked on the coordinate plane to come up with generalized “(x, y) ->…” rules for the different cases of the different transformations (reflections across x-axis, y-axis, y=x, y=-x; rotations about the origin clockwise, counter-clockwise, 90 deg, 180 deg)
Special Right Triangles
45-45-90 and 30-60-90
Special because one is isosceles and the other is half of an equilateral (or equiangular) triangle.
Key idea: All isosceles right triangles (45-45-90) are similar by AA similarity.
All 30-60-90 triangles are similar by AA similarity.
We let the smallest side of each triangle equal length 1 for easy reference.
By the properties of each triangle, we can determine one other side automatically:
- For a 45-45-90, since it is isosceles, if one leg is 1 unit, the other leg is 1 unit.
- For a 30-60-90, since it is half of an equilateral triangle, the smallest side is half of the largest side.
In both cases, we can use the Pythagorean Theorem to determine the 3rd side in each reference triangle.
From there, any time you see a 30-60-90 or 45-45-90 triangle with one side given, you can use the two reference triangles discussed above (with smallest side equal to 1), to determine scale factors, create proportions or use other identifiable patterns to solve for missing sides.
Simplest Radical Form
When dealing with the √2 and √3 sides of the reference triangle, it is important to always leave your answer in simplest radical form (no perfect squares can be left as factors inside the square root sign). Useful rules to remember:
- √a•√b = √(a•b) Ex: √2√3=√(2•3)=√6
- √(a•b)=√a•√b Ex: √50=√(25•2)=√25√2=5√2
- Rationalizing the denominator: If you have a/√b, you want to multiply the top and bottom of the fraction by √b, so that way it becomes (a√b)/b. Don’t forget to check if you can simplify the integer fraction a/b.