Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.



Wow this is quite vague.... In my humble interpretation, this is referring to the geometric mean relationship and to special right triangles. These are two relationships are founded in similarity of right triangles and lead nicely into trigonometry. I am sure that there are other items that fit in this category but I am going to focus on these two for now.






(1) The student will be able to derive the three geometric mean relationships.


(2) The student will be able to use the geometric mean to solve for sides of triangles.


(3) The student will be to construct the special right triangles.


(4) The student will be use exact values in solve for sides and angles in the special right triangles.








The underpinning concept of similarity unlocks many new powerful relationships such as the geometric mean and special right triangles. Once we understand that two angles locks in the side proportionality of a triangle then we are able to solve for lots of different things.





Students need a way to remember the three geometric means. One teacher told me that where the side touches the hypotenuse connects to the other two pieces that get multiplied. So in the case of a side, it touches the full hypotenuse and the piece on its side of the hypotenuse or when the altitude touches the hypotenuse it touches the two pieces. This is an easy way to remember the relationships but it doesn't address the bigger issue which is they don't understand it.


The big issue with special right triangles is when divide by or . Students struggle with this greatly. They think that if 8is the hypotenuse that we are 'dropping' off the giving us sides of 8. Yes the sides are 8 but we didn't drop it off, we divided it. When the hypotenuse doesn't have in its value students struggle to know what to do, even though the 'know' the relationship.




The connection is similarity. At the heart of the geometric means is the similarity of the three triangles found in that special situation. This is also true about special right triangles, the power of the special right triangle is that it works for all triangles with those angles .... similarity.



These relationships all connect to the concepts found in trigonometry. Trigonometry is the relationship between angles and sides in right triangles. The exact values of the special right triangles are early representations of the trigonometric relationships.





MY REFLECTIONS (over line l)

Geometric mean is always a difficult area for students to grasp. I have heard a number of teaching techniques to help students remember it... I just wish I could address more than just remembering it.. I want them to understand it. I still believe pulling out the triangles and revisiting the similar triangles is the best way. That is why in this area I start with that very activity.