A review of Chapter 9

In 9.1 we learned "The Theorem of Pythagoras."

Volcabulary review: A theorem is a mathematical statement that must be true because is can be shown to be true through deductive reasoning starting with axioms (or conjectures) which are true because they are so obviously true. We can't prove an axiom, we just accept them as true because we have to start somewhere. A good example of an axiom is: "the shortest distance between two points is a straight line." So a theorem is only as true as the axioms it rests upon. But theorems, unlike axioms, are usually not obvious at all.

Pythagoras was a man that lived in Greece around 500 BCE. (BCE means "before the common era." 1 CE [or common era] was the year many believe Jesus was born).

So the Pythagorean Theorem states that for any right triangles, the square of the distance of the two legs will equal the square of the distance of the hypotenuse. Or a^2 + b^2 = c^2.





Remember that a right triangle is a triangle with a 90 degree angle in it. (Seen left.) The longest side of a right triangle will be the side opposite the 90 degrees, in the picture it is called a. This longest side is called the hypotenuse. The shorter sides are called the legs.




Can anyone post an example of the three ways we proved this to be true to the class last week?


Problem 1 on page 465 can be solved easily once you fully understand the theorem: 13*13 is 169 and 5*5 is 25 so 169-25 is 144 so a=12 because 12*12 is 144.

But problem 2 on page 465 uses addition rather than subtraction because in this case you don't get the hypotenuse: 12*12 is 144. 15*15 is 225. 144+225 is 379 and the square root of 379 is about 19.2. So c is 19.2.

So remember it is an "addition" problem if you do not have the hypotenuse. Otherwise it is a "subtraction" problem.

The rest of the problems on page 465 are variation on problems one and two. If you do not understand how to do problems 3 or 4, do not start the lesson for the day until you do. I will try to provide you with individualize tutoring. Please identify another student who doesn't get it and pair up with them.


In 9.2 we learn the converse of the Pythagorean Theorem which states that "If the lengths of the three sides of a triangle satifsy the Pythagorean equation, then the triangle is a right triangle. See page 469. Any group that desires to do the investigation on page 468 is welcome to do that instead of starting with the lesson for today. It shouldn't take too long.

On page 473 we learned how to use radical expressions. Anyone who still struggles with these should carefully read page 473 and do problems 1 through 20 on page 474. Pair up with someone else who also struggles with this and I will give you individual help during class.


In 9.3 we learned about two special right triangles that it would be well for everyone to try an memorize. They are the 30-60-90 right triangle and the 45-45-90 triangle. Anyone who wants to post examples of these should do so. Anyone who feels they would be better off doing the Investigation on page 475 and 476 instead of starting with the lesson for today should do so.