okay my last video is on measurement I just wanted to talk about probably the
most useful theorem in woodworking or construction and that's a Pythagorean
theorem so let me just put this over here and there's two ways you could use
the Pythagorean theorem so the Pythagorean theorem named after
Pythagoras. first off it only works in right triangles so the
way you denote a right triangle here's Pythagorean theorem the way you
show right triangles you put that little box in there so this says it is a right
triangle that angle is exactly 90 degrees and the Pythagorean theorem will
work okay so just a couple things about triangles first you use a capital letter
for measure of an angle lowercase letter for the length of a side opposite that
angle okay so if this is side a it is opposite angle a this is side B opposite
angle B and then this is side see in every right triangle
there are two legs in this case a and B and it's common to use a and B for the
legs and then there's one hypotenuse and the hypotenuse is always opposite the
right angle I'm going to draw a little box that's a 90 degrees so the
hypotenuse is always opposite that right angle and it's common to call that right
angle C with the hypotenuse lowercase e okay so the Pythagorean theorem says if
one leg squared plus the other leg squared is equal to the hypotenuse
squared so you could see why this is so useful because everything could usually
split into a right triangle on a jobsite even if you have some crazy truss you
know and all sorts of webbing and you try to find out part you could always
drop an altitude down and create a right triangle and find missing pieces if you
have two of those pieces so this Pythagorean theorem you either
the length of the two legs you can find the hypotenuse or if you have the
hypotenuse and one leg you can find the other leg and if you only have an angle
and a leg or angle and a side you could use right triangle trigonometry to find
the other parts of that triangle so we'll put that off for another video
right now I'm just talking about the Pythagorean theorem so let's say in this
case right here I know that this altitude is three this leg right here is
four I have a right triangle I want to find C the hypotenuse I do one leg
squared plus the other leg squared equals C squared so three squared 3
times 3 is 9 plus 4 times 4 is 16 equals C squared 16 plus 9 25 equals C squared
get rid of that square I take square root of both sides and C is equal to
either 5 or negative 5 but I don't keep that negative because we're talking
about the Lent so that's how you use the Pythagorean theorem to find the
hypotenuse this one worked out well because they were all integers whole
numbers but I could use a calculator if I want and it'll work out just the same
so let's say let me just make one up that's not gonna work out well and I do
you like using measurements to find these I think these are pretty small but
let's say this is 1 right here this is what do you do not have one leg and the
other leg so I'm gonna say 1 squared plus 2 squared equals C squared 1
squared 1 times 1 is 1 2 times 2 is 4 5 is equal to C squared
get rid of that I take square root of it C is equal to the square root of 5 and I
don't know what that is so I gotta use my calculator and on my calculator I
just go here's the square root button can you see that there's a square root
button in blue so I'm gonna go second square root of 5 and I get to point to 4
so C is equal a to point 2 4 that's pretty close to 2-minute quarter inches
let me check and see if that works yeah that is pretty close to 2 and a quarter
inches there you see that there so that's how the Pythagorean works
Pythagorean theorem works really useful I go over one more example let's say
instead now I'm giving a hypotenuse of three and a leg of two what is my other
leg so now I have a leg hypotenuse I'm looking for another leg I like to label
it a B C side opposite a is a lowercase a so that's what I'm looking for a so I
have a squared plus two squared equals hypotenuse squared 3 squared a squared
plus four equals nine oh I didn't mean to do that it's gonna work out they
squared equals subtract four from both sides so a squared equals five square
root of both sides and a is equal to square root of five and I do that on my
calculator just to you in a quarter so you could use the Pythagorean theorem
given two legs to find the hypotenuse or give it a leg a hypotenuse find the
other leg and again the Pythagorean theorem is probably the most useful
equation or theorem used in carpentry and I'll show you how it's also used so
if this is true if it's a right triangle one leg squared plus the other leg
squared equals hypotenuse the converse of that is true too so the converse is
going to say if one leg squared plus the other leg square equals hypotenuse
squared then it's a right triangle so this is how you check if things are
square right so again let me run through that again
the Pythagorean theorem says in a right triangle one leg squared plus the LX
squared equals my pot knees right but the converse says if one leg squared
plus the other leg squared equals hypotenuse then it is a right triangle
and you've actually done this before I'm sure on a job site most people call it
three four five Inc so let's say I have a house and I want my fence to be
perpendicular to my house the way I lay that out is I measure down the house
four feet I measure down the fence three feet and then I pull a string tight but
I pull a string tight when that string is exactly five feet then it's a right
angle so what I'm saying here it's 3 squared plus 4 squared equals 5 squared
then it is a right angle and if you need even more distance all of those ratios
will work to you so let's say if I didn't use three instead I used a nine
so if I measured down building a deck if I measure down nine feet times four by
three twelve feet and five by three fifteen so if I measure down here
exactly nine feet better down here exactly twelve feet a pull of string
when that string is exactly 15 feet it'll give me a right angle so this
Pythagorean theorem is really cool because it's so useful if I know it's a
right triangle then the Pythagorean theorem works or if the Pythagorean
theorem works then it is a right triangle so hopefully that'll help you
in the woodshop and out on the job site it is a useful theorem all right keep
practicing it to figure it out