I want to show you another formula that
is basically just another name for the
"Pythagorean Theorem". And it's called the...
Distance Formula. Now what makes the distance formula
special is that it can find the
distance between any two points.
Let's take two points. We'll use the point (4,3). Let's see...
over 4 and up 3 right? And the point...
(-1, -3). so we'll go back 1 and...
down 3. And we want the
distance between those two points. I want
to know that exact distance; that red
segment (if you would). And we'll call that
distance (for now) ...x. And since we've gone
to the trouble (which took a long time
I might add)... of drawing a graph grid we
hopefully, if you look hard, can see
that there is a right triangle with that
side x as the hypotenuse and we can
easily since it's on a graph measure the
legs. One of them is 6 and one of them
is 5. Take the time and go ahead and
measure the units there's six units on
the right and five units on the bottom.
This means we can use the Pythagorean
Theorem. You may remember is... 6 squared
(one leg squared) plus 5 squared (the other
leg squared) equals... It doesn't
equal the hypotenuse... equals the
hypotenuse squared.
Now we can calculate what these numbers
are... 6 squared is 36. 5 squared is 25. and
they're supposed to add up to x squared.
Well they do add up to 61. So if 61 is...
x squared, the way we would find X would be...
to take the square root of both sides to
get X by itself. And that would make X in
this case, equal to the square root of 61.
and if we used our calculator we could
get a decimal estimate which would be
about 7.81
which seems reasonable. What I mean
is if, looking at the picture, if I got
78.1... I'd know something's wrong.
ok... Well we didn't learn anything
new here. We just used the Pythagorean
Theorem because we went to the trouble
of drawing the graph. But can we do
it without a graph? Same points now...
Consider... i'll bring the graph back (just
for reference)... the two x-coordinates of
these two points... If i subtract them...
4 minus -1 ...remember what happens
when you subtract a negative... 4 minus
-1 ...we'll get 5. Remember where
that was? That was the bottom leg.
That was the distance between the
two x values. And if we did the same
thing with the two Y values... The y part of
the first point minus the y part of the second point...
or 3 minus -3 we'll get... the
other leg. The difference between the
y-values is the Y side which is, of
course, 6. And we can then use the
Pythagorean Theorem... 6 squared plus
5 squared (same thing we did before) equals...
x squared. We calculate the multiplications,
add them up, take the square root of both
sides and get the same answer (without
drawing the graph)... that's the point. Now
what we just did is the "distance
formula" as crazy as it may seem
it's the difference between the
x-coordinates of the two points; x1 minus
x2, and then squared it and that's what
we did...
added to... the difference between the y
coordinates and then squared it, and then
at the very end ...remember what we did, we
took the square root. That nightmare (what
you're looking at) is the distance
formula. And it exists to find the
distance between any two points
without drawing a graph. Now you could
always sketch a graph if you get
confused and use of a Pythagorean Theorem
because that's just what this is just
another form of the Pythagorean Theorem.
Ok let's do one more. This time we'll do
it BLIND. We're not going to be able to
see the graph. We're going to find the
distance between the points (-2, 4)
and (2, -1) using this distance
formula. That's what it's for.
Well let's see... We subtract the x
coordinates... negative 2
minus 2... and I hope you know that is...
-4 Now you're worried aren't
you because you see a negative. But hold on...
Let's subtract the y-coordinates... 4 minus
-1 ...and in this case ...
... We get a 5. Now that negative is
going to get taken care of because recall
that we're going to square these
answers. -4 (which is what the x1
minus x2 is... squared, we will add to
that... the 5 squared... and look what we
get ... 16
Hi Mom! positive 16 added to 25, and if we
add them up... we get 41. And don't forget
We need to take the square root, So the
resulting hypotenuse (or distance if you
would) would be... the square root of 41
which is about 6.4 Again it's in the
range to seem reasonable and I can prove
to you in fact that it is. Just in case
you don't believe that, we can do
the Pythagorean Theorem (using
this formula) without a graph... let's do the
same two points WITH a graph... and
hopefully we'll get the same answer.
(-2, 4)... going back 2 and up 4...
and (2, -1) going over 2 and down 1
that distance we'll call X. And now that
we have the graph...
we can actually count the legs. The left
leg is, as you count, five boxes. We got it
by subtraction earlier. But with the
picture, if we go to the trouble of
drawing the picture we can actually
count the side... and the bottom is
son-of-a-gun... it is 4 Again, before we
used the distance formula and just
subtracted and got it. But with the picture
we can verify in fact that it is 4.
and if we plug these into the
Pythagorean Theorem... 4 squared plus...
5 squared equals that hypotenuse squared
...son-of-a-gun, we get the same numbers and
the same sum of 41 (being x squared)
Of course, don't forget, we have to take the
square root of both sides and... yet again
we get that x is... the square root of 41
Or... 6.4 So if you have a graph go ahead
and draw your points and use the Pythagorean Thereom
... but if you don't want to go to
that trouble you can just use the
distance formula and get it in that way.
OK... Dig in with that homework and don't forget to
write down the distance formula in your
notes !