Geometry Question Made a Maths Teacher SWEAT

hey guys what you're about to watch is my live walkthrough of a fairly tricky geometry question that I came across when I did the grade n tough paper 2 that I uploaded about a week ago now I decided to do this because I realized that watching an hour and a half an hour and 40-minute video might not be viable for some of you so I decided to snip out this question to then present to you in another format but let me know if this is something that's useful should I select individual questions and then show you how I kind of work through them or is the kind of long paper format better for you I will link the full paper in the description anyways in case you guys want to have a look but without further Ado I will let past me uh go through the question there's always one question that just looks like it's going to give you a headache so there are two touching C and it's always to do with circles anyways Circle one and circle two um it's fairly clear that this over here is circle one and this is circle two um gives us the radi or radius whatever you wa know the radi so five and four that's how I know this was Circle one because the radius was bigger which means it has to be the bigger Circle um C is the center of circle to uh obviously AB lies on the diameter of circle one yeah AB is going through the diameter AB is also perpendicular to a line passing through the center okay this is actually a deceptively useful piece of information AC and BC are tangents to Circle 1 and path through Center C AC and BC tangents yeah cool co co cool okay yep I've got it um find the area of the triangle ABC so this is a big triangle now that does look really complicated and almost always in questions like this almost always you're going to be using the fact that well a couple of things the first one is this this line made me this this line yeah this line made me pretty happy because what they're trying to give you a hint and it's very rare that the exam board are nice and merciful but they're trying to tell you to do well okay it's a bit lower than that to do that and and why is that so you think why would they give me that information well this goes straight from the center of one circle to the center of the other right and they're touching so this part of this is the radius of the first Circle that's 5 cm this part is another 4 cm so this Total Line is actually 9 cm long so we have our first number now if you remember how to do the area of a triangle Le you need the perpendicular height and then the base so we've actually worked out one of the two things we need so just as a reminder um I know this sounds really sarcastic but it's half base time height but remember the height has to be perpendicular so we've actually already gotten one of the two things that we need super super useful Okay so we've used that information and it proved very fruitful indeed um now the next thing thing I look at is this bear in mind a lot of this information we have used right like I have technically used the radio both of these the fact that it's the center I have technically used because I said that this line was going through the centers of both circles so all of this like we have used it technically speaking um but those last two are really interesting so what is important about tangents when you're talking about circles well they're perpendicular right so uh God that was terrible um yeah okay let's let's undo that and actually give it a better go there we go right angle and a right angle and what another thing that we can do this is pretty exciting um the reason why I wanted to do that is we need to work out the coordinates of point A and B which as you can see are the Y intercepts of both of these graphs right so it's the Y intercept Y intercept once we work that out working out the base is Trivial we just need to find the difference in the two coordinates and we're done um which is easy that's not too bad but we do not to work out the equation of these lines we need the gradients which we we can and can't work out uh because we do we we okay we know that they're perpendicular so we know it's the negative reciprocal but we don't have any coordinates which we do need to work out the equation of the line anyway so I need to think about what we can do with the information we have just before we go on uh it actually might be a good idea for me to label some of these points so I'm actually thinking of two different ways of doing this one way is to just work out this base here and then and do it from there there's another way I'm thinking of um which I'm going to keep it to myself for now actually no I'm not the other way is because we've split this directly in half we could just work out the area of the top triangle and then work out the area and then just double it so the question is which one's more useful well the fact that the perpendicular is going to help in a second and because we need to use some trigonometry or Pythagoras to work out some of the other some of the other information that we need because we don't have enough information at the moment simply put so if I work out okay so because we couldn't use the fact that um we can't use the Y intercepts by using the equation of a straight line there is another way we could work out the the length of this line what I'm going to do is if I look at this angle here right that angle is in this triangle but it's also in this triangle both of which are perpendicular so what I'm going to do I'm going to do something very clever I'm going to work out the the angle using this inside Triangle d c that angle will be the exact same as a d c and what I'll be able to do is work out the length a d which will be the same as the length of DB that's probably the best way we can do it okay so question is what is this red length well if you remember it's just going to be the radius of the circle so it's actually going to be 5 cm so if I want to work out that angle Theta it is opposite over hypotenuse so that means it will be s theta equals opposite so 5 over hypotenuse 9 Theta = inverse s of 5 9 let's use our calculator uh change it back to the normal setting inverse S 5 over 9 gives me 33 I'm just call it 33.7 de I wonder if to do a separate video on this question this is a really good question once I've done that I'm now going to get rid of this very carefully okay I'm going to leave I'm not going to push it now I have Theta so I'm going to oh I'm going to get rid of that it's 33.7 and I want the side a d so let's just call it I no X again well that's fine right it's opposite over adjacent it's tan So Tan 33 2.7 equals opposite over adjacent so X over 9 cuz again this is still perpendicular I really think I should do a video on this one this is a really good question which equals uh well I mean I'm going to work it out on the calculator not going to do it in my head 6.01 do I want to keep it super accurate let's keep it super accurate let's do 6.01 cm uh to 3sf which means the area of a DC is half base time height actually no I should label it I'm getting I'm getting too excited all right 6.01 uh which is going to be half 6.01 * 9 and that means the area of ABC is two times that entire thing beautiful so 0.5 time answer * 9 take that double it 54.1 cm squ to 3sf what a question pretty good yeah that a pretty interesting question but again it's using the same kind of skills that we've used before and yeah and as you can see all I did is I drew all of the information out and then and then made a decision the hardest part there was seeing that there were two triangles that used the same angle that that I can see that being a bit of a problem because we don't really use that fact very much in GCSE maths so you won't be used to using that fact