Math Matters: Why Do I Need To Know This?
Bruce Kessler, Department of Mathematics
Western Kentucky University
Episode Eight
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Similar and congruent triangles – Hard-to-measure
distances
Objective: To illustrate the concepts of similar and congruent triangles can be
applied to real-world problems, particularly in problems that involve a distance
that is physically difficult to measure.
Hello, and welcome to Math Matters: Why do I need to know this? If you’re new to the
program, let me tell you what we’re doing here. I’m attempting to take the mathematics that
you find in entry level math courses in general math, college algebra, and even the courses
for elementary school teachers and show you how that mathematics applies to everyday living
and I’ve got some nice examples to show you this morning, so let’s get after it. The first
thing I’d like to talk to you about is an application of just plain old geometry. I’m not going
to do any trigonometry with you or anything like that, but I’d like to use some geometry to
show you how to measure hard to measure distances. Things like trees or solid things that
you can’t just simply run a tape through, things like that.
I’m going to do all of this now without getting into sins and cosines. The kind of things
I’m going to do today, we’ve got civil engineering students up in Ogden college that could
tell me exactly what these measurements are by using some trigonometry and using some
very specialized equipment to get distances and angles and things like that. I’m not doing
anything that sophisticated. I’m doing something that you and I, you could do if, you know,
you had to know how tall a tree was because you don’t want it to fall on the house. You’re
getting ready to cut a tree down and you don’t want it to fall on the house. Do I have room
to fall this tree? That’s a good question. You don’t want to take chances on your house, so
let me give you the set-up here, and I need to lay down some ground rules.
So I’m going to talk about two different types of triangles today. The first is congruent
triangles. Now that’s simply, that’s a big word, that just simply means that they have the
same shape and size. So here are a couple of congruent triangles. Now I’ve changed the
orientation of them, but they do have the same shape. The same shape is indicated by the
fact that they have three sets of corresponding angles which I’ve indicated here with one arc.
This angle corresponds to that one, this one corresponds to this one, this one corresponds to
this one, and this one corresponds to that one. That forces them to have the same shape, but
the same size occurs when they have corresponding sides that match up, “b” to “b” and “c”
to “c”. Now luckily for us, there are six bits of information there that I don’t typically need
all six bits of those information to show that I have congruent triangles. (Figure 1)
I can typically get by with just three. For example, if all three sides matched up – this
side corresponds to that one, this one to this one, this one to this one. Then those triangles
have to be congruent. That’s the dreaded side side side theorem. (Figure 1) I can also get by
with side angle side, two sides and the included angle which this side corresponds to that one
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Figure 1, Segment 1
and this one to that one and then the angle in between those matches up. That forces the two
triangles to be congruent. (Figure 2) Vice versa, I could take two angles and the included
side, that forces these triangles to match up. (Figure 3) Or I could even have two angles
and a non-included side. The two angles here and a side that’s out from between those two
angles. That’s enough information to force congruent triangles. (Figure 4)
Now I’ll show you how having congruent triangles can help you measure distances. The
example I’m going to show you is, I’m going to measure the height of a tree near Thompson
Complex Central Wing. In fact, it’s right in front of Science & Technology Hall and what
I’d like to do is measure the height of that tree using no specialized equipment. Now here’s
an old boy scout trick. I want to thank John Spraker for showing me this, it’s a nice little
idea. This is the back of my head, and take some item like, let me have you up here for just
a second so I can demonstrate this. You take something like a pen or a ruler or whatever
you’ve got, a fixed length and you hold this up to the item so that it matches the base and
the top. And you may have to back up or you may have to scoot forward, whatever. That
marks actually a triangle. Now we’ll go back to this. Here’s me holding up my little pen or
whatever I have here. That gives me a triangle: the top of this, the bottom of this, and the
height. (Figure 5)
Now, and let me bring you back up here and show you this, it’s better if I illustrate it up
here first. Once you’ve got this, then take your pen and lie it with, lay it with the terrain. It
doesn’t have to be a right angle, but just lay it with the terrain, okay? And then you’ve got it
with the base of the tree here and then you’ll mark a point out there, kind of this equidistance,
but out there in space and then you can go measure that distance, and we’ll go back to this.
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Figure 2, Segment 1
Figure 3, Segment 1
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Figure 4, Segment 1
Figure 5, Segment 1
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You’ve got a point in space to measure to and that gives you the distance because what I
have here are congruent triangles. This angle and this angle will match up thanks to my
unit of measure here. This angle and this angle will match up as long as I do things kind
of equidistant across here. I want to kind of do things in the plane that’s running kind of
parallel to this point through the tree. And then I have a side here that’s shared. So I’ve got
the dreaded angle side angle condition and that gives me congruent triangles and that means
that that distance right there is going to be the same as that distance right there. (Figure 5)
So here’s my example of doing that. I took, I couldn’t find a ruler, that’s a sad statement
for a math teacher, but I couldn’t find my ruler. All I could find was my clipboard. So I took
my clipboard out there and I pulled this down with the height and then I took my clipboard
and I laid it sideways and it marks a point out here just a bit past this light post that I’m
going to measure to. And I’m trying to do it so that this and this would be about the same
distance apart. And I measured that to be about 70 feet. (Figures 6 and 7)
Figure 6, Segment 1
So that’s a decent estimate I think, now I don’t have anything else to check it with, but I
can check it with similar, there’s another trick for similar triangles. These are similar. They
have the same shape that means that they have the same angles that match up, but not the
three sides. But it turns out I only need two of the angles to get this because the sum of the
angles inside of there is always 180 degrees. If two of them match up, the third one has to
match up. They will have proportional sides. Whatever the ratio of lengths are from “a” to
“x”, these are corresponding sides will be the same as from “b” to “y” and the same as with
“c” to “z”. (Figure 8)
So the shadow trick, as I like to call it, with tall trees, now you’ve got to do this when the
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Figure 7, Segment 1
Figure 8, Segment 1
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sun is shining which I’ve had a little trouble with here lately, but I caught a few moments of
sunshine. What you do is you go measure the shadow this thing casts. That’s this distance
right here and then you take a yardstick or something, something with a fixed length and you
set it up on the same terrain and you measure that shadow. Now the angle that the sun is
hitting the both of these is the same and by laying these on the same types of terrain, those
angles are the same, so I have similar triangles and I can form this proportion and then
solve for the “h”. I’ll know “a” and “b”, I can measure “x”, I know “a”, just solve for “h”.
(Figure 9)
Figure 9, Segment 1
So real quickly, I had a couple of grad students from the math department helping me
out with this. They measured the shadow. I actually had one standing in the road, that
probably wasn’t a good idea and here’s a long distance picture of the two of them measuring.
(Figure 10) I forgot to get a picture of them using their yardstick, but they used a yardstick.
(Figure 11) They measured the length of the tree’s shadow to be 68 feet. Their yardstick was
three feet and the yardstick shadow was two feet, ten inches which is 17
feet, you’ve got to do
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a little bit of conversion there. My feet units cancel out, I do the cross multiply trick, multiply
by the reciprocal to solve for “h,” and it turns out to be 72 feet which is pretty consistent
with the other measure I had. So I feel pretty good, pretty confident that that tree is about
70 to 72 feet tall. No surveying equipment, no surveying equipment used. (Figure 12)
I flashed some things through there pretty quick so let me flash a little bit up there slowly
so you can absorb it a little bit.
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Figure 10, Segment 1
Figure 11, Segment 1
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Figure 12, Segment 1
Summary page 1, Segment 1
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Summary page 2, Segment 1
Summary page 3, Segment 1
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Summary page 4, Segment 1
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Different-base numerals – Understanding your computer
Objective: To illustrate the importance of understanding different-base numeral
systems as a means of understanding the inner workings of digital equipment.
The next thing I would like to show today is not so much something you need to know
for everyday uses but it’s something you just simply need to know. We live in a digital age,
and everything that we do pretty much is based on something I am going to show you, a little
piece of mathematics I am going to show you. Unfortunately, it is a piece of mathematics
that most people have no concept of whatsoever. So, I want to take just a moment to show
you this, not so much for the application standpoint, but so that you will understand that
this is used around us all the time, okay?
I want to talk about different base numerals. And what I mean by that – I, I’ve got to
define the word numeral so, here’s a we call this a number typically. One hundred eightynine; but that it’s both actually. When I use the term numeral I’m referring to the symbol
that represents that amount, and we have some kind of feel for that amount because we have
lived a while and we learned how much that is. When we say number we are actually referring
to the amount, not the symbol, but the actual amount. Now, the numeral up here that I am
showing you that is a base-ten numeral. And what I mean by that is . . . it is based on powers
of ten, each of the places in this numeral represents powers of tens; ten to the zero which
is one, ten to the one which is ten, ten squared which is one hundred. So what this actual
means if you literally break this down – it means that I have one group of one hundred, plus
eight groups of size ten, plus nine singles. Okay, and that gives me the one hundred and
eighty-nine. (Figure 1)
You might ask why is it ten. Well, you know, right here you go, right? All I’ve got to
do is wiggle these things and you know luckily for us our ancestors I guess had ten fingers,
we have ten fingers, and this is the number system that has evolved. If we were like the
Simpsons, I don’t know if you ever watch that show or not, but they have four fingers on
each side, we’d have a different number system.
Alright, now, here’s a here’s kind of the rough idea. Things that we use everyday, like
computers, DVD players, iPods, all these things that we seem to love and are dependent on
uh, uh game stations, all these kind of crazy things, they don’t understand base-ten numbers,
numerals. They understand something different because they only really have two fingers.
They have switches instead of fingers and switches that are either one or off. So, they have
a completely different numeral system. Well, not completely different, it’s similar to ours,
but it is not based on ten it is based on two. It’s called binary numbers and there’s only two
digits; zero which typically would mean the switch is off and one which means the switch is
on. I think I have those correct. If not, the computer science will correct me and I will fix it
next time. (Figure 2)
So inside your computer system, here’s the way they do here’s the way it will do math.
We have this string of ones and zeros and the place values now are not powers of ten but
they are powers of two. One, two, four, eight, sixteen, thirty-two, sixty-four, one hundred
twenty-eight. Now for us to understand that, it has to do a little conversion and print the
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Figure 1, Segment 2
Figure 2, Segment 2
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correct characters, but here’s what this number I’m sorry here’s what this numeral represents,
it means that I have one group of twenty-eight, I have one group of size thirty-two, I have
one group of size sixteen, one group of size eight, one of size four, none of size two, and one
of just single. And I think if I add all this up, let me do it real quick, 160, 180, 189. So,
that’s the same number, okay, it’s subtle here, but that’s the same number, just a different
numeral. Okay, and that’s the kind of numeral that computers, game boys, whatever it is,
that’s the numeral system that they work on. (Figure 2)
Okay, now you may think that requires a bunch of programming; actually it doesn’t. The
cool thing about this is that, even though we’re talking about using a different base numeral
system, all those algorithms that we learned back in grade school still work, even though we
have a different base number system numeral system. For example, if I wanted to add a
couple of numbers, 82 and 35, okay, I think we can do this in our heads. So 117. 117, I’m
good with that answer. Let me convert these over to base-two. So, 82 would be a group of size
sixty-four and then a group of size sixteen, which I think, let’s see, let me do the arithmetic
here, sixty-four, that would be eighteen, here’s group sixteen, so I got two left, yep. So, that
is correct. And then 35, well, a group of size thirty-two and I’ve got three left, so two and
one. Okay, if you add these using the old algorithms, okay, and keep in mind that when I add
one and one I get two, but “two” but two in base-two would be one-zero. So, zero and one
is one, one and one is one-zero, I’ve got to carry the one, one, zero, one, one, one. When
you convert this back, this is my group of sixty-four, group thirty-two, group of sixteen, no
eights, a four, no twos, and a one. That adds up to 117. That’s the correct answer – all
the algorithms still work. So, there’s nothing new algorithm-wise on our computers, it’s just
that the number system is the numeral system is different. (Figure 3)
Figure 3, Segment 2
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One more example of that, so you don’t think it just works with addition. If I multiply,
okay, 16 times 5, for example, gives me 80. Let me do the conversion – sixteen would be one
group of sixteen, nothing left over. Five is a four with a single. Remember how you multiply
these, one times all that gives me this and one times all this, I’ve got to move it now to right
here. And add and this gives me a sixty-four plus a sixteen which is eighty. It’s really slick
how all that works out. (Figure 4)
Figure 4, Segment 2
Did you ever notice this one; here’s a cool one for you. Did you ever notice that numbers
are divisible by nine if you can add their digits up and that sum is divisible by nine? Hopefully
someone has pointed that out to you. So like, one hundred eighty-nine is divisible by nine
because one plus eight plus nine is eighteen and that is divisible by nine. But, there is
nothing really magic about the nine it’s just that it is one less than the base-ten. So if
I asked a question like well is it divisible by seven. That’s hard; we don’t have a lot of
divisibility things for seven. But, I can convert to base-eight. Okay, now that requires me
to do group things in this fashion, one singletons, and then eight, and eight squared. That
would convert into two, subtract two sixty-four on the left equals sixty-one, seven times eight
goes into sixty-one and left with five. And then, if I add up those digits, two plus seven plus
five is fourteen and that’s divisible by seven. Which, yeah, that number is divisible by seven.
So, that’s a nice little way to do some divisibility things. (Figure 5)
And real quickly I want to show you this, there are other bases that are used periodically
by computer programmers. They use a base-sixteen system called hexadecimal. And what
they have to do is replace these double things in base-ten with single digits. Now, what that
gets us into, and the reason they do that, is that when they dump their binary stuff, it is a
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Figure 5, Segment 2
mess, but they can go through and clump it by groups of four and then get the values for each
of those groups of four in binary and then write in terms of hexadecimal. So, the fourteen
would be an E and the eleven is a B and so forth. The cool thing is that this number and
this number are the same. (Figure 6)
That’s where it applies, it’s not that you will necessarily do that much of it but you need
to understand that this is going on in all that digital equipment that you have. Alright, let’s
flash up some summery pages of the information I just showed you.
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Figure 6, Segment 2
Summary page 1, Segment 2
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Summary page 2, Segment 2
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Piecewise functions – Modeling non-continuous situations
Objective: To show how piecewise functions are used in real-world problems by
giving an example involving postal rates. The concept of a piecewise function is
introduced, as well as inverses and composite functions, although not by name.
The last thing I have to show you today is fairly algebraic. This is the material we’re
covering in college algebra. There’s a topic in college algebra where we, part of what we do
there is graph piecewise functions and kids hate this because you can’t hardly do it on your
calculator. I mean you can do it on your calculator, but you have to know how to do it on
your calculator. You have to be pretty adept at using your calculator to pull this off. And
it’s confusing to students and they don’t really understand why we need to do this anyway
because they can’t think of a good use for it. I want to show you a good use for it, okay?
First, I should tell you exactly what I’m talking about. A piecewise function has different
output rules depending on which values you put in. So for values in this case less than “a”,
I would use this rule. And for values between “a” and “b”, actually including “a” and “b”, I
would use “r2 ,” this rule. And then for “x” greater than “b”, I would use this rule. But, kids
get confused because they graph this thing across; you know they’re used to graphing things
across and all the way across. They’re only defined for certain portions of the real number
line of the x-axis. So it takes a little bit of practice to get used to graphing those on your
calculator. (Figure 1)
Figure 1, Segment 3
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The example I would like to show you involves the U.S. postal service. They distinguish
between packages, large packages, and oversized packages. The way they do that is they take a
look at two values of the box that you’re sending. The length which is the longest dimension,
which in this picture it would be the up and down dimension. And then they measure the
girth which is the perimeter of a cross section. This runs perpendicular to the length. So if
the length is this way, the cross-section would be this way and you measure that perimeter.
And they use a thing called length plus girth, if the, and you’re just adding them together.
So, if the length plus girth is less than 84, then there’s a certain rate they will apply. If it’s
between 84 and 108, they call it, this isn’t just a package, this is large package. There’s a
different rate. And then this is oversized package and it’s at an even higher rate. And I’ve,
I just made these numbers up. These aren’t US postal rate things, but these numbers are
actually the accurate numbers. (Figure 2)
Figure 2, Segment 3
Now what I’d like to do is put myself in this situation where I have to determine if this
is possible. I mean if you’re doing this, say you have a business and you have to send things
to folks. You don’t have your own trucks to deliver things, you have to mail them, then this
might be something you have to worry about. I want to determine the cost per package to
ship a package of weight “w” pounds. So “w” in this case is my weight, and I’m looking for
information here about what would be the ideal weight that my packages should be so that I
don’t waste a lot of money here. The whole idea of being in business is to make money. So,
I want to do things in a nice efficient, costly, cost-saving manner. (Figure 3)
Okay, so I’m going to make some assumptions to make this a little easier. I’m going to
assume we’re using a square package. That’s not a wild assumption. That’d probably be the
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Figure 3, Segment 3
one to minimize, maximize volume but minimize the surface area and that’s probably easy to
find. That automatically tells me then what my lpg is, it’s the perimeter which is 4x plus the
length x, so that’s just 5x. It also tells me the volume which is just x3 . All these dimensions
are the same, I’m calling it x. So, that gives me a little bit of information right there. I’m
also going to assume that things are, the weight of the things I’m putting in these boxes is
about one ounce per cubic inch. That’s not terribly, terribly heavy, but it could be, it could
be fairly heavy. That will enable me to then calculate the weight in terms of this x. Here’s
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of a pound, so it’s
the volume and then for each cubic inch of volume I have, an ounce is 16
just that formula. And I’m also assuming that there’s a flat fee per package. You know, no
matter what it weighs, they’re going to charge you this this $2.90. (Figure 4)
The cost then for sending a package per weight w, okay, you take the rate times the weight
and that rate depends on the weight, so that’s not multiplication, that’s function notation.
And then you’re going to add $2.90. Now here is the issue: this thing, this “r”, this rate, I
don’t have it in terms of “w”. I have it in terms of lpg and I’m using these kind of cooked
up numbers. I do have the lpg in terms of x, I do have the weight in terms of x, so I need
to get this rate thing in terms of weight. So I’m going to bridge the gap by doing this. This
is invertible. I can say “w” in terms of “x” or I can say “x” in terms of “w”. Multiply both
sides by 16 and then take the cube root of both sides.
√ Then, I can take this x, put it back in
my lpg formula and that means that the lpg is 5 3 16w. Now I can take this and replace it
in this formula. (Figure 5)
The only thing I have to worry about is, I’m getting this kind of ugly business right here
in these inequalities and I would like to just have “w”. So I need to do a little solving. This is
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Figure 4, Segment 3
Figure 5, Segment 3
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messy, don’t confuse this, this is a big mistake, don’t confuse messy with difficult. “Messy” is
real life. The real world is messy. “Difficult” is, it means the theory behind it is harder, the
concept, or how to make it work is harder. We’re just doing some basic arithmetic here. It’s
not difficult, it’s just a little messy. Okay, let’s go back. I would need to solve this inequality
for “w” so I divide both sides by five. I would cube both sides, get this kind of ugly number
and then divide by 16. So that’s the right hand side of my first inequality. And then really,
you just kind of do that we all of these. The other is bounded by 108 and the other is bounded
by 130. So you get these bounds on “w” and I only did one side because these are repeated.
(Figure 6)
Figure 6, Segment 3
So I’m ready now to get the cost function. It’s weight times “r”, so I just took “w” and
multiplied it times the rate here and then plus the $2.90, the flat rate. And so this is now,
this cost function is a piecewise function which I’m going to graph for you, okay? If it gets
out past almost 1,100 pounds, then it’s getting too big to ship even as an oversized package,
so I have to stop. But you can notice that there’s some breaks in this map, in this graph.
And if you’re in this 630 range right here, then that’s a good indication that you need to be
careful because you can ship a package that weights 625 pounds at this rate, which is fairly
low, or you can ship something that weighs 630 pounds at a much higher rate up here. So,
if I’m trying to save money, right, it’s to my advantage to stay below this number and not
have to pay this extra, extra high fee right here. (Figure 7)
Good example of how you want to do this kind of analysis of your business practices to
try to save money and it involves piecewise functions. The things that I flashed up there were
pretty quick, so let me put a summary page up that talks about what I talked about.
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Figure 7, Segment 3
Summary page, Segment 3
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Closing
Well that’s our show for today. I hope you enjoyed the things we talked about. I’m always
trying to get things that are just interesting to the general public here. I do have to do things
kind of quick, so please know that you can find each of the episodes as downloadable files
from our web page and we’ll flash the address up in just a second. Also, please take a time
to contact me to let me know that I’m doing some good here, or if you’ve encountered some
mathematics and you think “How on earth could I ever use this stuff ?” I love a challenge
like that. I’d love to get a, take a shot at it. Or if you have a good application that you think
might make a good segment on the show, I’d like to hear about it. And the e-mail address
will be flashed up in just a few moments too. With that, I’m done for this week. We’ll see
you next week.
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