Looks like everybody is in their seats so
let's go ahead and get started...Welcome
it's sunny over here in California hope
everything is going on well wereabouts
you, excited to see so many here on
this workshop tonight we won't be taking
too much of your time or get right into
it we're going to get into the content
we're going to run through a few methods
of using the Distance Formula, go through
a practice problem and then i'll have
some additional resources and things I
would like you guys to jump into
following this quick workshop but
nonetheless let's go ahead and get
started so in this workshop again we
will explore the Distance Formula which
falls under the main topic of Straight
Lines which that also falls under the
Subject of Mathematics, now equations,
symbols, tables and information on the
various topics covered under Mathematics
can be referenced on pages 18 through 32
of the NCEES Supplied Reference Handbook
9.2 version for the Computer Based Testing
so what is the Distance Formula?
the Distance Formula is a general
restatement of the Pythagorean theorem
and is used to find the distance between
two given points either on a line or in
space now the formula is given by this
right here...d is equal to the square root
of y2 minus y1 squared plus x2 minus x1
squared where d is the distance between
the two points given by the coordinates
x1 and y1 and x2, y2...now the distance can
be expressed as unitless or in
a provided unit coordinate system
such as feet or meters...
it doesn't matter now x1 and x2 are the
x-coordinates of the respective points
and y1 and y2 are the y coordinates of
each respective point...now the
distance formula itself can
be referenced under straight lines on
page 18 of the NCEES Supplied Reference
Handbook, this again is 9.2 version for
the Computer Based Testing...now that we
have established the general definition
of the distance formula let's discuss
two common problem types that we may
encounter on the FE EXAM so the first
problem type we will be given the x and
y-coordinates of two points will give be
given the x1 the y1 and x2 y2, we will just
be given two points in this case it's
just a simple plug-and-play we put those
points into the distance formula
equation and we calculate it, there's not
much work that needs to be done however
there's a second scenario which is a
little bit more complicated
and takes a little bit more effort if
you're going to manually move forward
with calculating it, an d that's when we
are given the distance between two
points, we are actually given the distance
and we are given three of the four X-Y
coordinates with one of those
coordinates being unknown and we're
asked to solve for that unknown
coordinate so again the first case is a
simple plug-and-play scenario but the
second case requires calculations using
the quadratic formula now this approach
in the in the case that we use the
quadratic formula can be fairly time
consuming and it could open us up to you
know multiple calculation errors as we
make our way through our manual process
so in this calculator workshop what we
will do is will go over how we can solve
for problem that falls under that second
scenario where we have to find an
unknown coordinate in the distance
formula problem and will do that using
the TI-36X PRO and we will compare that
process against the typical manual
approach using the quadratic
equation so now we know that in order to
use the quadratic equation the first
thing that we must do is manipulate any
given equation to ensure that it is in
the standard quadratic form which is ax
squared plus b y plus c is equal to zero
we've all seen this equation this is the
standard quadratic equation once it's in
this  position...or in this form we can
then define the values for A, B and C and
plug them into the quadratic equation
which is X is equal to negative b plus
or minus the square root of b squared
minus 4ac divided by 2a so again this is
a very familiar and comfortable formula
that we've all seen throughout our time
in the university as well as after so
we're very comfortable with making these
calculations and our first go-to is to
probably solve these types of problems
using this quadratic equation but when
it comes to time and efficiency on the
exam
there's much faster ways which I'll show
you today so again the quadratic
equation itself can be referenced under
straight lines on page 18 of the NCEES
Supplied Reference Handbook again
version 9.2 for the Computer Based
Testing so now that we got all that
definition out of the way let's go ahead
and jump into a typical problem we
may see on the exam so the problem
states...given two points with the
coordinates (30,-3) and (a,7)
determine the value of a if the distance
between the points is square root of 424
so we're going to walk through this
problem first by going over how to solve
it manually and then we will run
through how to solve it using your NCEES
approved non graphing calculator
specifically the TI-36x PRO...now recall
that this problem falls under the second
problem type as described just
previously in a few slides prior where
we are given the distance and three
other for X-Y coordinates of the two
points we are now asked to solve for
that one unknown coordinate which is a
so to be clear in our in our end goal
this problem requires that we use the
distance formula to setup an
equation relating the given points and
the distance between them so that we can
solve for the value of the unknown
coordinate a so recall that the distance
formula is d is equal to the square root
of y2 minus y1 squared plus x2 minus x1
squared again where d is the distance
between the two points given by the
coordinates that we are given y1
and y2 are the y-coordinates of the two
points and x1 and x2 are the
x-coordinates of the two points we were
given now in this problem we are given
the distance of the square root of 424
were given the x1 coordinate which is 30
were given the y1 coordinate which is
-3
we're given the y2 to coordinate which
is 7 but we don't know the x2
coordinate which is given as a so the
first thing we need to do is plug
these values into the distance formula
so doing that we get the square root of
424 is equal to the square root of 7
minus -3 squared plus a minus
30 squared so
now that we have one equation and we
have one unknown variable which is a we
can now move forward with solving using
one of the two methods we've already
mentioned so the first technique is to
again simply solved by manually first
expanding and simplifying the
equation and then putting it into the
standard form of a quadratic equation we
can then use the quadratic formula to
solve for the unknown variable a and
then we got our second method which is
much more efficient is way faster and
that's to use the equation solver
function on your calculator the TI-36x PRO
so what we're gonna do now is
actually walk through method 1 which is
a very comfortable method for us it's
where we just manually do it we can do
this we're trained to do it but the name
of the FE EXAM is to be most efficient
so if your calculator can do it let's go
ahead and train ourselves to know how we
can do it on our calculator...so this
let's go through the manual
equation solving, that's method 1 so
starting with the distance equation we
formulated with the given information
again we have the equation as its stated
right there, our second step is to simply
just square each side to get rid of the
square root not a big deal
the next step...step 3 will just
go ahead and simplify and then for step 4
we will go ahead and use our FOIL
technique to distribute the binomials
recall that the letters FOIL stand for
First, Outer, Inner, Last...so FIRST
means you multiply the terms which occur
First in each binomial and then
Outer means you multiply the outermost
terms in the product Inner means
multiply the innermost two terms
Last means multiply the terms which
occur last in each binomial then finally
we simply take the product and combine
any like terms which may occur so
following this technique we're going to
get what you see here on the screen 424
is equal to 10 squared plus a large
equation right there
so for Step 5 we are going to go ahead
and simplify that in step 6 we're going
to combine the like terms and then in
step 7 we're actually going to
combine a little further so the next
step is we just need to rearrange and put
this equation into the standard
quadratic form again the standard form
is ax squared plus b y plus C so doing
some quick math we can go ahead and you
know subtract 424 from each side, do all of
our quick calculations and we come up
with 0 is equal to a squared minus 60a
plus 576, so now that the equation is in
standard form we can now define our
values for A, B, and C which is obviously
1 for A, B is -60 and C is 576
so now that we have all the bar A, B, and C
terms we can simply just refer to the
quadratic equation which you probably
have memorized and we just plug in our
values we go through a couple
calculations we know since the value of
a is squared here in the standard form
we know that there's going to be two
roots...so we can't forget that
so we just put our A, B, and C into the
quadratic formula we solve for and we
get two roots one being the x1 root is
48
x2 root is 12...therefore the two points
that are a square root of 424 away from
the point (30,-3) from our problem
statement would be (48,7) and (12,7)
so I just cruise through that
problem really quick you've already saw
on all the graphics were already here on
the screen but if we were to do this by
hand it would obviously take a little
bit more time then you know me just
illustrating it already done so we
obviously don't want to spend all of our
time manually doing these equations in
these formulas that can be done on our
calculator...you know we have limited time
on this exam we only have six hours a
hundred and ten questions to get through
so when we come across the simple
problem like the distance formula we
need to just take and use our calculator
and use it to its the best
ability so what I want to now show you
is actually method 2, and that's using
your NCEES Approved TI-36x PRO
calculator so again we're going to just
start off with our standard equation so
in either method we always have to
formulate that distance formula first so
we have to have that information
now the next step what we're going to do
is actually initiate the numerical
solver on TI-36x PRO, so we're gonna
hit the second button and we're going
locate the sine and inverse sine button
which you can see right above that it
says num-solv, so that is your
numerical solver that's how you initiate
your numerical solver
now step 3 once you initiate that
numerical solver the screen will display
to empty squares with an equal sign
between them
now you can enter the equation into the
solver and to simplify everything I went
ahead and just in my head as you can all
we can all do is we just square both
sides we got rid of the square root you
can input that into the numerical solver
if you want but you don't need to
because it just makes it more
simple so the equation that we put in is
424 is equal to 7 minus -3
squared plus X minus 30 squared now
the TI-36x PRO doesn't have a so for
the variable you hit the X button which
is on the left side of the calculator
and i'll be showing you here in a second
a video of this process so you can see
where all these buttons are located but
you hit that variable button and you're
also going to need to square some things
so you have to hit that x squared button
which is also on the calculator...now once
you're complete you hit that enter
button this will submit the equation and
bring you to a new screen on this screen
we will need to provide a starting value
of iterations for the equation solver to
iterate through and search for a value
that satisfies the inputted
equation now this screen should say
something like enter and solve on one
line x is equal to some numerical value
and then the third line can be solved
for X so one thing I do want to note
again there's going to be two roots for
this equation for the specific equation
so the calculator is only going to give
you one root at a time so in order
to solve for multiple roots of an
equation we can iterate through
intervals of 100 to ensure we are
changing the starting point of the
iteration process so to start we will
scroll down to the line that says where
it says numerical value and we're going
to input the value of zero we're going
to scroll then down to the next line
solve x
and we're going to press enter now
depending on how far away the solution
is from 0 it may take a few seconds for
the calculation to complete and the next
screen to load but nonetheless once
it does load the screen will read like
this, the first line will say solution
second line will be x is equal to 12
third is going to be L minus R equals 0
and then the fourth line is going to be
solve again or quit so we now know that
12 is one of the values for our unknown
coordinate a, now real quickly for
everybody here's a strategy for the exam
okay we have one root so a good test
strategy at this point would be to give
the choices look at the options of the
answer see what answers are available
for the problem and just give them a
quick glimpse if any of those choices or
answers do not contain 12 as an
x-coordinate then we know we can
eliminate that choice as an option maybe
at this point there will only be one
option that has 12 as an x-coordinate in
that case that's your answer right there
and we don't have to move to iterating
the next solution so since we know there
are two roots again for this equation as
the X term is squared we need to solve
now for the second value of a so at this
point the calculator will be blinking on
the solve again at the bottom of the
screen so once you are ready to do move
forward with the next solution just
simply press enter you'll be
brought to the same screen as you were
previously where the first line will be
where the first line will be...
step 4, sorry, where the first line will be
enter and solve, the second will be x is
equal some numerical value and that's
actually going to say x equals 0 because
that's what you previously put in and
then the third line is going to be
solve X so this time around you're
going to put 100 in for the numerical value
switch down to solve and push enter on
solve X, so once you do that you'll get
another solution, and that solution this
time should be x is equal to 48 so we
now know that X equals 48 is
the second value for our unknown coordinate
a now at this point we can evaluate our
answer choices and look for the answer
choices that define the two points that
we've now determined again those two
points are going to be (48, 7) and (12, 7)
exactly what we got previously using our
manual process so now what I want to do
we are now 21 minutes now into this
tutorial i want to go to the next screen
and this is a video of me actually
solving this problem so it's going to
show you kinda in real time how fast
this can be done so i'll be i obviously
for illustration purposes took it nice
and slow but here is actually what it
will look like...
[VIDEO OF SOLVING PROBLEM WITH THE NCEES APPROVED CALCULATOR TI-36X PRO]
and that's it...I stepped through that really
slowly so you could actually watch my
fingers hit the buttons but it's
literally that fast if we have an
equation we can define that variable
within the numerical solver we can then
iterate from certain values of X and
find the solution that fast...so that's
going to conclude today's webinar review
calculator tutorial on the distance
formula and what I want you guys to do
next is actually hop over to the
PREPINEER Q&A FORUMS where we will be
providing additional practice problems
were actually going to put together a
nice little package for you guys to take
away
we're going to provide you the
step-by-step procedures to complete
these types of problems as well and then
we're also going to create or initiate a
conversation around the distance formula
so if you guys have any questions about
this particular function on your
calculator the TI-36X PRO, go ahead and
hop over there, myself, Daniel and all the
other professional engineering mentors
will be there to answer any questions
you may have so for now that's it for
today be looking out for the next
calculator tutorial we're always taking
suggestions send any any ideas of ones
you may want to see in the future...
again there's a couple tools that the
NCEES gives to you on the exam, one being
the calculator, two being the NCEES
Reference Manual...if you end up becoming
a master at both of these the exam is
going to become much much easier easier
you can be solving things much more
efficiently...you're not going
to run out of time so for now just want to say
May God Bless, have a great weekend
thank you for taking some time out of
your Friday to spend with me and watch me
go through a calculated tutorial and
until we talk again take care bye-bye
