Maths Teacher Makes Geometry Question Easy

hey guys we have another difficult problem solving question however I feel like this one has more obvious steps but the problem is it gets really really hectic because the equations that we're going to make are very very long uh to put simply so this is also going to be a bit about laying your workout properly again I have had very very intelligent students very capable students who can solve pretty much anything but because they're working out is laid out messily they make small mistakes and that's what I'm going to go through here is exactly how I would personally lay this out and how we can you know compare these two things because I wouldn't write this all on one line because you're going to see that this is actually a pretty hefty equation so we have two shapes here we have this part of a sector it's not really a full sector and we have a circle and they're telling us the area of shape A and B are the same so as you can imagine working out the area of shape a and I like to write out in words first so if I have area of shape a we're going to have the area of a sector minus the area of the small sector right so area big sector minus small sector and as you can imagine that might be a somewhat large equation and for B we're going to just have the area of the circle and those two things should be identical so that's the first thing I like to do just so I can kind of focus myself a little bit then I ask myself how do I work out the area of a sector now I personally don't like memorizing stuff so instead I think to myself well what is a sector a sector is just a fraction of a big circle so if I have the area of a sector it means I've got a fraction of a big circle now where does that fraction come from it comes from the angle that the sector subtends so how wide the sector is so for example if it's 180° it's actually a semicircle it's half a circle if it's 90° it's a quarter of a circle and so on so what you're going to do is say what fraction of the circle this is so in this case it's 45° out of 360 and then you can simplify that a bit later this is a non- calculated question by the way and if you want the area of the big sector well that's going to be pi and then times the radius squar now the radius is actually going to be these two things added together because don't forget we're talking about the big sector the overall pizza slice here so we have 3x - 1 plus + 2x that would be 5x - 1^ 2 and then take away the same angle so 45 360 pi and in this case the radius is just my 3x - 1^ 2 and that should be equal to the area of the circle which is p piun r 2 but remember they've given us the diameter so I'm going to half this I get 1 - x^2 from here there's a ton of different things you can actually do uh this here is actually the difference of two squares if you want to kind of factorize it but in my opinion the easiest thing to do is first of all divide through by pi and then we're going to simplify 45 over 360 um if you half 360 a few times you will get to 45 or you can just kind of work it out normally however you want to do it so we could divide top and bottom by five for instance in order to get 9 over 72 and then of course we can divide it by um n right afterwards so we get basically an eighth so what we have here is we have an eighth of 5x - 1^ 2 minus again an eth of 3x -1 2 = 1 - x^2 I might want to rewrite that as - x + 1 2 just because I'm going to expand brackets soon but before I expand brackets and again this is something I see some students do is they expand it even with a fraction in front the problem there is again dealing with fractions is quite tricky if you have like 5/8 minus 1 + 23 it's going to get really tricky so instead I would times both sides by eight because I personally hate fractions so 5x - 1^ 2 - 3x - 1 2 again that is the difference of two squares I don't think um factorizing it would make it much easier so I'm not going to bother expanding the left side we get 25 x^ 2 cuz it's 5x * 5x then we get 5x * -1 so - 5x and we have that twice that' be - 10 x then we have plus one the next mistake I see is when you expand a bracket that has a minus in front a lot of people would write Min - 9 x^2 and then write the rest of it personally I write it still with a bracket because I want to make sure that I don't make any mistakes again this is all about laying your workout neatly and following through very very carefully because if I get make a mistake on any of these numbers I get the wrong answer so again we have 9 x^2 CU it's 3x * 3x then we have - 3x minus another 3x for the two uh cross terms and then plus one and again now if you look here the signs all change because of the negative over here I'm going to do the exact same thing so we have - x * - x which of course is x^2 then we're going to have 1 * - x twice so it's going to be - 2x and then we're going to have + 1 so now what we're going to do is do 25 x^2 - 9 x^2 which gives me 16 x^2 then we have - 10 x - - 6 x so that's - 4x then we have 1 - 1 which is nothing then the other side we have 8 * x^2 so we have 8 x s 8 * - 2x is - 16 x 8 * 1 is eight then I'm going to move everything over to the left hand side yeah I think the left hand side would make sense so minus 8x s from both sides gives me 8 x^2 then I'm going to add 16 x to both sides so that gives me+ 12 x then- 8 from both sides zero I'm going to divide everything by four to make my equation as simple as human human humanly possible Z and then obviously we just factorizing for the value of x so we're looking for two numbers where should I do this I might do this off to the side here so we're looking for two numbers that times to AC so that be times to -4 and then you need to add to three so that would be -4 and 1 so that means we have 2x^2 - 4x + x - 2 = 0 so factorize again split it down the middle factorize the first two 2X x - 2 then we have just plus one so our two equ sorry our two brackets our 2 x + 1 and x - 2 = 0 so of course we get two answers we get X is- a half and X is 2 but if you notice in the question it does say work at the value singular of X well if I have a look at this what does X represent X represents a length for example 2x as an example well 2 * - half is -1 does that make any sense can I have a length of minus one no so we're actually taking the positive value because if I sub that into all of the different values of X it works okay it makes sense so X must be equal to two there we go so again if you take it very carefully and you lay your work out relatively neatly like I've done it will make a lot more sense leaving the negative sign outside the bracket again this is a really common mistake even grade eight and N students make okay you have to lay your workout in such a way that you it requires the least amount of brain power to read because that helps you avoid mistakes in the future the only thing I probably would say is I probably would have written 45 over 360 off to the side and simplified it just to be super super sure that I got it right but other than that it should be all okay so again hope you found that useful again think of the exam technique more than the actual math I'm sure the math isn't too bad