2. Families 30 60, 90 45, 90 Resource Geometry/ Sophomores

3. When working with right triangles, there are certain set of groups that you must know. There are three main sets that you must learn: a 3-4-5 triangle, 5- 12-13 triangle and 7-24-25 triangle. The larger of the number of each set corresponds to the hypotenuse. Then the other two are part of each set correspond to the legs of the triangle. Another important note is that other groups of sets can be similar to the three basics ones presented here. For example a 3-4-5 is similar to a 6-8-10 triangle because of simplifying ratios between the two triangles. It is important for you to remember these basic families because if will save you time rather than doing a Pythagorean theorem to solve for each side if two sides are known. Example

4. This triangle is similar to which basic family? A. B. C.

5. Another special triangle is the 30-60-90 degree triangle. Each degree corresponds with its own leg. For instance the 30 degree angle corresponds with the leg across from it which is x. The 60 degree triangle belongs to the leg across from it, x square root of 3. And finally the 90 degree, right angle side, corresponds to the 2x leg (hypotenuse). Placing a value in for x will change the length of the sides of the triangles. For example if x=2, the length of the leg across from 30 degrees is 2, the leg across from 60 degrees is 2 square root of 3, and the hypotenuse is 4. Example

6. If x=3, what are the lengths of a 30 60, 90 triangle? A.B.C.

7. Finally, a different special triangle is the 45-45-90 degree triangle. Each degree corresponds with its own leg. For instance the 45 degree angle corresponds with the two leg across from it which is x. And finally the 90 degree, right angle side, corresponds to the x square root of 2 leg (hypotenuse). Placing a value in for x will change the length of the sides of the triangles. For example if x=2, the length of the legs across from the 45 degrees is 2, and the hypotenuse is 2 square root of 2. Depending of the value of x, can change the lengths of the sides. The legs across the 45 degree are congruent because the legs and angles equal each other. Example

8.  If x=4, what are the side lengths of a 45-45- 90 degree triangle? A.B.C.

9.  Image of 3,4,5 triangle on slide 1 is from: http://justcolleges.com/tests/triangles-tutorial-for-sat- gre-test-preparation/  Image of 30-60-90 degree triangle on slide 1 is from: http://hotmath.com/hotmath_help/spanish/topics/30-60-90-triangles.html  Image of 45-45-90 degree triangle on slide 1 is from: http://www.sparknotes.com/testprep/books/newsat/chapter20section4.rhtml  Image of 3,4,5 right triangle on slide 2 from: http://mathyear2013.blogspot.com/2013/08/another-rule-about-pythagorean-triples.html  Information from slides 2, 4, 6 is from: http://www.sparknotes.com/testprep/books/newsat/chapter20section4.rhtml  Images of 5-12-13 & 7-24-25 triangle is my own creation.  Image of 30-60-90 degree triangle is my own creation.  Image of 45-45-90 degree triangle is my own creation.

11. Make sure you are simplifying as low as possible First, choose two legs to work with. example: 15 & 36 Second, make them into a fraction. example: 15/36 Third, simplify the fraction. example:15/36=5/12 =

12. Remember that the leg across from the 30 is x, the leg across from the 60 is x square root of 3, and the leg across the 90, which is also the hypotenuse, is 2x

13.  Remember that the leg across from the 45 is x, the leg across from the other 45 is x, and the leg across the 90, which is also the hypotenuse, is x square root of 2.

15. =