This Blog exists for the collective benefit of all geometry students. While the posts are specific to Mr. Chamberlain's class, any and all "geometricians" are welcome. The more specific your question (including your own attempts to answer it) the better.
One of my favorite students indicated a concern over the topic of geometric mean... so let's discuss for a bit...
Similar to many other words in the English language, the word "mean" can have many meanings (pardon the pun) in mathematics. Basically, it is a "measure of central tendency." Even the fairly specific term "geometric mean" can be applied in different contexts, such as the geometric mean of a data set. This serves the purpose of making you miserable and confused, which brings joy to the math gods and nasty math teachers like me.
That said, what's gonna be on the test??
Before we start, let's agree that in the proportion a/b = c/d, a&d are the extremes and b&c are the means... unhappy people? No? Good.
Next, In a proportion, IF b&c are equal we DEFINE that situation by saying that b (or c) is the geometric mean of a&d. For example:
1/2=2/4; so 2 is the geometric mean of 1&4 4/8=8/16; so 8 is the geometric mean of 4 and 16 2/8=8/32; so 8 is (also) the geometric mean of 2 and 16. 4/radical48=radical48/12; so radical48 (or 4*rad3) is the geometric mean of 4 and 12
Another way to define GEOMETRIC MEAN is this... The geometric mean is the square root of the product of two real numbers (say that three times fast!).
Unhappy people? No? Good!
(Waaaaa!!!! but what do I need to know for the test?????)
So what happens when we draw an altitude from the right angle of a right triangle to the the hypotenuse? (suggestion to readers: draw a rt. triangle and drop an altitude... NOW!). We KNOW that we create two "mini-me" similar triangles. We know this because: 1) we are very smart 2) each of those new mini-me triangles shares an angle with the big-momma triangle AND they also both have a right angle. That means that the corresponding third angles MUST be congruent based on the AA~ postulate... happy? yes? good!
SOOOOOOOOOOOOOOOOO... look at the altitude you drew. It is the longer leg of the smaller mini-me triangle and the shorter leg of the larger mini-me triangle... agree? yes? good!
So the shorter leg is to the longer leg (of the smaller mini-me triangle) as the shorter leg is to the longer leg (of the larger mini-me triangle).
So, as stated in the Theorem 8-1 corollary 1:
When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse (cut by the altitude).
If you play around with corollary 2, you should be able to describe this relationship using similar logic... I'll leave that to you.
Please let me know if this helped. You may need to read this a couple of times, but reading alone will not make the learning "stick"... this will help only you if you draw a few examples and try a few problems.
hmmm... you should be able to see that since the longer leg on the 30-60-90 is 9*rad3, the shorter leg must be 9, right? Then the hypotenuse is 18, yes?
Then if we look from the "south" the 45-45-90 says that x*rad2=18. Hopefully, you can solve for x from there? You'll have to rationalize the denominator, of course!
hi, number 29 i got 16 and (16 radical 3)/3 i dont understand how thry got just 16 radical 3. i was supposed to post this a couple hous ago, but my computer decided to be a complete jerk and crash on me. also on the self test. how do you do number 4? yes, sorry im late. but again i must say technology hates me!
Yeah I was stumped with this at first but then i drew a picture and found it gives you the legs of a right triangle and then it's just Pythagorean theorem.
i got a different answer from the book for 2b and 2c anddd 5 on the self test..so how do you do those ?
ReplyDeleteohhkay i got it... nevermind :)
ReplyDeleteOne of my favorite students indicated a concern over the topic of geometric mean... so let's discuss for a bit...
ReplyDeleteSimilar to many other words in the English language, the word "mean" can have many meanings (pardon the pun) in mathematics. Basically, it is a "measure of central tendency." Even the fairly specific term "geometric mean" can be applied in different contexts, such as the geometric mean of a data set. This serves the purpose of making you miserable and confused, which brings joy to the math gods and nasty math teachers like me.
That said, what's gonna be on the test??
Before we start, let's agree that in the proportion a/b = c/d, a&d are the extremes and b&c are the means... unhappy people? No? Good.
Next, In a proportion, IF b&c are equal we DEFINE that situation by saying that b (or c) is the geometric mean of a&d. For example:
1/2=2/4; so 2 is the geometric mean of 1&4
4/8=8/16; so 8 is the geometric mean of 4 and 16
2/8=8/32; so 8 is (also) the geometric mean of 2 and 16.
4/radical48=radical48/12; so radical48 (or 4*rad3) is the geometric mean of 4 and 12
Another way to define GEOMETRIC MEAN is this... The geometric mean is the square root of the product of two real numbers (say that three times fast!).
Unhappy people? No? Good!
(Waaaaa!!!! but what do I need to know for the test?????)
So what happens when we draw an altitude from the right angle of a right triangle to the the hypotenuse? (suggestion to readers: draw a rt. triangle and drop an altitude... NOW!). We KNOW that we create two "mini-me" similar triangles. We know this because:
1) we are very smart
2) each of those new mini-me triangles shares an angle with the big-momma triangle AND they also both have a right angle. That means that the corresponding third angles MUST be congruent based on the AA~ postulate... happy? yes? good!
SOOOOOOOOOOOOOOOOO... look at the altitude you drew. It is the longer leg of the smaller mini-me triangle and the shorter leg of the larger mini-me triangle... agree? yes? good!
So the shorter leg is to the longer leg (of the smaller mini-me triangle) as the shorter leg is to the longer leg (of the larger mini-me triangle).
So, as stated in the Theorem 8-1 corollary 1:
When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse (cut by the altitude).
If you play around with corollary 2, you should be able to describe this relationship using similar logic... I'll leave that to you.
Please let me know if this helped. You may need to read this a couple of times, but reading alone will not make the learning "stick"... this will help only you if you draw a few examples and try a few problems.
Good Luck!!
that did help! thanks Missa C!
ReplyDeletep 301 #9?
ReplyDelete301, #9, eh?
ReplyDeletehmmm... you should be able to see that since the longer leg on the 30-60-90 is 9*rad3, the shorter leg must be 9, right? Then the hypotenuse is 18, yes?
Then if we look from the "south" the 45-45-90 says that x*rad2=18. Hopefully, you can solve for x from there? You'll have to rationalize the denominator, of course!
That's Mistah C. to you, Dr. J...
thanks missa c! how about you post a test answer.....?
ReplyDeleteI agree with jamie. post the test answers if all possible
ReplyDeleteword
ReplyDeletehi, number 29 i got 16 and (16 radical 3)/3
ReplyDeletei dont understand how thry got just 16 radical 3. i was supposed to post this a couple hous ago, but my computer decided to be a complete jerk and crash on me. also on the self test. how do you do number 4?
yes, sorry im late. but again i must say
technology hates me!
Yeah I was stumped with this at first but then i drew a picture and found it gives you the legs of a right triangle and then it's just Pythagorean theorem.
ReplyDeleteOHHHH!
ReplyDeletethanks a bunch nora!
you guys are just SOOOOOOOO SMART!!
ReplyDelete#5 i just dont know but i think it is e. also # 14 i think it is 2 and 3 but thats not an answer
ReplyDeleteon the quiz what was the answer to #13 on the back?
ReplyDeleteThe answer to #13 was 12.
ReplyDeleteI believe I graded that incorrectly on many of your quizzes.
The re-take is tomorrow during 1st period before we go to the courthouse.
that explains so much. thanks!
ReplyDelete