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we're on problem 135 I just did some Pilates so I'm slightly out of breath if X is less than zero is y greater than zero so if X is less than zero is y y greater than zero all right statement number one x over y is less than zero x over y is less than zero so this is actually pretty useful right they're telling us that X is a negative number right so we have a negative here and we're dividing it by some other number and they're telling us that that num that the resulting number is still negative so we have a negative divided by something is equal to a negative so I think you learned when you learn negative numbers if if Y was negative well if Y was zero this would be undefined and if Y is negative if this was a negative then the resulting when you divide a negative by negative Y would have a positive so Y has to be a positive number and it can't be zero so this actually tells us statement 1 alone tells us that Y is greater than zero just based on the fact that you have a negative that it's just telling you that a negative divided by Y is still negative right this whole thing is still negative so Y has to be a positive statement number two tells us Y minus X is greater than zero and that just tells us that Y the C add X to both sides that tells us that Y is greater than X well that's not enough it says if X is zero is y greater than zero X could be you know maybe X is equal to minus 10 and Y is equal to minus 9 so in this case why wouldn't be greater than zero or maybe X is equal to minus 10 and Y is equal to plus 9 both of these would satisfy this condition so we don't know we this doesn't tell us any information on whether Y is greater than zero is greater than zero so statement one alone is sufficient and statement two doesn't do much for us problem 135 no I already did 135 136 136 okay they've drawn a circle there so I guess I'll draw a circle as well so that's the circle and they have C they have a center the center goes up like that and then it's like that and there's a right angle right here they call this oh they call this X they call this Z and then actually connect those two points they connect those two points and then they call this at this point right here that's why let's see what they're asking what is the circumference of the circle above with Center us it is one another circumference so if we could figure out a radius then we know circumference is 2 pi times the radius and we'd be all set so let's see what they what information they give us statement number one the perimeter of triangle o X Z is 20 plus 10 square roots of 2 perimeter is equal to 20 plus 10 square roots of 2 it's interesting what do we know about this well we know this is going to be an isosceles triangle we know that if if this length right here is X that this length is also going to be X because this is a circle so all the circle is the radius is constant right and each of these sides are the radius of the circle so those are both going to be X and then what is what is this going to be it's going to be I thought we just know from Pythagorean theorem x squared plus x squared is going to be equal to that side squared it's going to be let's call that C squared so you get 2x squared is equal to C squared or C and I'm calling this side right here C is equal to the square root of 2x times the square root of 2 whatever X is right so the parameter the percent of sorry the perimeter of this entire triangle is going to be 2 X's right X plus X so it's going to be 2x plus X square roots of 2 X square roots of 2 and statement one told us that this perimeter is equal to this right here is going to be equal to 20 plus 10 square roots of 2 well I think you could do a little pattern matching here and solve for X I mean do we have a this is actually even though it might not look completely like it this is a linear equation and you can solve for it and and immediately that should be of statement one is sufficient but you can just do a little pattern matching and say well this works when X is equal to 10 so actually we are able to figure out what X is equal to and then as I said before X is the radius of the circle and then the circumference of the circle which is the whole question is about that's 2 pi times the radius circumference is 2 pi times the radius we just figure out the radius is 10 so it equals 20 pi so statement one is sufficient what a statement to do a statement to the length of Arc XYZ is 5 PI X Y Z is 5 pi so they're telling us that this length is 5 pi but that is 5 PI well do we know what proportion that is of the entire circumference well sure because they tell us that this is 90 degrees right here right this is 90 degrees of 360 degrees of a circle so this the arc length right here is exactly going to be 1/4 of the circle and we know that because 90 degrees is 1/4 of 360 so 5 pi is going to be equal to is going to be equal to 1/4 times the circumference or that the circumference is going to be equal to 20 pi so each statement independently statement 1 or statement 2 independently is sufficient to answer this question the circumference of the circle 137 what is the value of ay to the fourth minus B to the fourth e to the fourth minus B to the fourth and immediately just even looking I mean one thing I'm tempted to do is rewrite this as a squared squared minus B squared squared because I was just I just glanced the statement wanted the head square it's there so I'm gonna do that right and you know that a squared let me do different letters you know that x squared minus y squared is equal to X plus y times X minus y right so here X is a squared and Y is B squared so this is going to be equal to this is just the original 8 of this is going to be equal to a to the squared a squared plus B squared times a squared minus B squared and then we could simplify this again so this is going to be equal to a squared plus B squared we could simplify this as a plus B times a minus B I think that's going to be helpful just glancing at this two statements that they gave us so statement one tells us that a squared a squared minus B squared is a squared minus B squared is equal to is equal to 16 so that tells us that just this is equal to 16 that is equal to 16 that doesn't help me much because I don't know what a squared plus B squared is going to be it just told me the difference between these two squares doesn't tell me anyway I don't know what their integers or anything statement 2 tells me that a plus B is equal to 8 and once again that doesn't help me tremendously by itself right if I just know that a plus B is equal to 8 I don't know what a minus B is I don't know when all of this is but when I use them together this is interesting right because we know that a plus B a plus B times a minus B is equal to a squared minus B squared well they just told us that a plus B is equal to 8 a plus B is equal to 8 so 8 times a minus B is going to be equal to a squared minus B squared statement one told us with 16 so that actually tells us that a minus B is equal to two so now we have two equations with two unknowns and we could solve for a and B and if we can solve for a and B then we can definitely tell you what a to the fourth minus B to the fourth is and just to show you that let me do it so let's see a minus B is equal to two add these equations to a is equal to 10 a is equal to five and then if evil to five B is equal to three so our original statement a to the fourth is five to the fourth five to the fourth minus three to the fourth so whatever that's what 625 minus 81 I think it doesn't matter you can figure it out so that both statements combined are sufficient to answer this question see you in the next video