hello um so triangles one very special triangle that you need to know about on the gmat um can't
be derived from a square and so we're gonna start here with a square and to make a triangle
it's very simple um we can divide it from a to d and now what we have is two triangles and so
actually i'm going to make this easier and just get rid of this part of it so we can just focus on
the triangle and since we're dealing with a square i know that each angle is 90 degrees
so that's going to be 90 and then we just divided an angle on half here and here so now
this is going to be 45 degrees and this is going to be
45 degrees so now we have something really interesting here um two things of note one
since this was a square both sides are equal so this side is equal to this side
and the this angle is equal to this angle that is the definition of an isosceles triangle
now this is a very special isosceles triangle because it was derived from
a square and thus a 45 45 90 degree triangle so that's what they're talking about when they say 45
45 90 are the angles 45 45 and 90. i'll write that right here great now what we want to do is
figure out um which we often have to do one of the sides so let's say in this square that
this side was one and this side was one just to keep things simple and we can figure out
the side a d using the pythagorean theorem does anybody remember the pythagorean theorem
i know somebody out there does i can hear you oh yes yes a squared plus b squared equals c squared
very well done so um a and the b that is our side and c is going to be the hypotenuse which is this
and so we're just going to plug in what we have we have 1 squared plus 1 squared equals
and here i'll just put the side a d a d squared so now 1 squared is 1 and 1 squared is 1 and 1
plus 1 is 2. i know it's tough math stick with me and so we have 2 equals ad squared and then
to get rid of the square root we need to uh or excuse me to get rid of the square we need to
take the square root so then we end up with a d equals square root 2. awesome so now we know
uh this side and so another thing that we know about an isosceles triangle that is a 45 45 90
degree triangle is that the two sides potentially could equal one and then the hypotenuse is square
root two but we can take this and generalize it and say that if um 1 is not 1 but let's say it's x
that would make this x and the only thing we need to do over here is put an x right there and then
we have a very easy way to deal with the sides and hypotenuse of this 45-45-90 degree triangle
so um it's important to know this aspect the sides and the hypotenuse so x x and x square root two
um but it's also important to remember how to get here a lot of the math on the gmat rewards you for
knowing how to derive things and how to manipulate uh different shapes and different formulas
it doesn't reward students who just sit there and memorize the formula because a lot of the
harder questions at the higher level of the test are really testing your ability to manipulate
these mathematical concepts and not just see if you can plug in numbers into a formula
so hopefully this helps you understand uh this special triangle and make sure that you can derive
this information on your own and not just memorize a formula and i think sherrod might understand it
a little bit better and maybe some people out there so if you have any questions feel free
to ask we're very happy to answer any questions you might have be excellent to the universe