for this question these are all the questions that we're going to do so there's seven of them but I'm gonna split the questions up onto different slides but if you want to just have a look at all the questions so long there they are and then I'll be splitting them up like that and like that okay so on to two different pages so here we go in the diagram C is a point on the y-axis such that a is that b is that c is oh no they don't give a c and then D is that they are a parallelogram okay we might need to know that so let's just highlight that one um K is the point that and that and L is a point such that that is parallel to that okay so everything that they've said is already shown on the diagram first question calculate the length of the diagonal DB okay so we need the length formula which is the same as the distance formula which is this one over here so the way it works is you can choose which one is point number two and which one's Point number one I'm going to choose this is point number one and this one is point number I just said the wrong thing this one's Point number two and this is point number one it doesn't matter though like you can choose it however you want you will get to the same answer okay so the length of DB is then going to be so it says that I must take the x value of Point number two which is a minus four then the x value of Point number one which is a four then the Y value of Point number two and then the Y value of Point number one which is a minus four see so if there's a minus four I just add it next to that minus it's okay and then I just go type all of this on the calculator exactly as we've written it over there ah we get a nice answer of 10. and in the test you must also just show them um the formula here so write down the formula like that and then in the next step you would show them the the values that you are using the next question calculate the coordinates of M which is the midpoint of DB okay so if we're to connect D to B There is some random little point which is the midpoint okay so for that we would need to use the midpoint formula okay so I don't know why but this formula sheet was a bit weird but there's supposed to be an equal sign over there so the midpoint is equal to that okay so once again we can just use Point number one point number two and so we could say uh four and then plus minus four over two and then minus 4 plus 2 over 2. okay so then we can go work it all out so we can you're going to end up here with a zero okay and then minus one so m is a point somewhere over here which has coordinates of zero and minus one the next one says calculate the gradients of a d okay so the gradient of 80. so for that you use the gradient formula so we're doing that between A and D so I'll just call this point number two and I'll call this point number one and so to work out the gradient you're going to say the Y value of Point number two which is a 2 and the Y value of Point number one which is a four and then the x value of Point number two and then the x value of Point number one like that go ahead you can type that all in on the calculator just like that and you should end up with a half now remember these questions do carry on okay so what we have already calculated in previous questions was that the length of DB was 10 we found that the midpoint of DB was 0 and -1 and we found that the gradients of a d was a half now the next one 3.4 prove that a d which is this one is perpendicular to a b okay that's easy remember how do we prove so how to prove that two lines are perpendicular you get the gradients get the gradients of each line okay then you multiply the gradients multiply the gradients and then if the answer is equal to minus one then we can say perpendicular so if the answer is -1 then they are perpendicular if it's not minus one then it's not perpendicular okay so we already worked out the gradient of a D in the previous slide okay we said that that gradient was a half so we now need to get the gradient of a B okay and so there we have the gradient formula over there okay so we're going to do the gradient now of a b okay so the gradient so what we can do is we can say that that's Point number two for example and that's Point number one it doesn't really matter as I said you can do the other way around so what the formula tells us that we need the Y value of Point number two which is a four then the Y value of Point number one which is a negative four then the x value of Point number two and then the x value of Point number one and if we had to go work that out we get eight over minus four which is minus two okay so remember what we said to C of two lines are perpendicular we work out the gradient of each one okay then we go and multiply them so let's go multiply a half which is this one's gradient and well let's first say here that we are multiplying the gradient of a d multiplied by the gradient of a b and so that's going to be equal to a half multiplied by minus two which is minus one and so there we have it because we get minus one we can say therefore a d is perpendicular to a b so the next question says give a reason why parallelogram ABCD is a rectangle so some of you reading that are like um what the hell like it's a parallelogram dude why is it now how's it also a rectangle but what we must remember and they they sometimes don't make this very clear in school like I didn't know this when I was in school but um a rectangle a rhombus a um Square well that doesn't really look like a square but a square all of these are also parallelograms okay but now they're saying why is the parallelogram a rectangle so so that's a parallelogram right but if you make one of these angles 90 degrees then all of a sudden it looks like that okay that then it's a rectangle so a rectangle a rectangle is actually a parallelogram with a 90 degree angle with a 90 degree angle so we've just proved that these two lines over here are perpendicular so that means that they are forming a right angle so the reason that it's a rectangle is that it is a parallelogram with an internal angle equal to 90 degrees so it's a parallelogram with an internal angle equal to 90 degrees now some of you might be like yeah but Kevin must we show that all of them are 90 degrees but guys remember it's already a parallelogram so what do we know about parallelograms we know that the opposite angles are equal and we also know that these are parallel lines so if you had to use co-interia you would know that this is already 90 and then if you had to use opposite angles again it's already 90. so you only need one angle to be 90 because the rest will all be 90 as well because of the properties of a parallelogram all right the next question determine the equation of k l okay so here's KL in the form y equals to MX plus C okay so we have y equals to m x plus C now m is the gradient now we have a bit of a problem because you can't get the gradient of ko because we don't know the coordinates of L however we do know that this line is parallel to this line which is parallel to this line so all three of those lines are parallel and when lines are parallel then they have the same gradient okay so that's that's how we'll get the gradient of KL so the gradient of KL is going to be equal to the gradient of a d why because they parallel so or let's say here that a d is parallel to KL and so the gradient of KL is a half so therefore we can say that Y is equal to a half X plus C and so now to find the point or the point of C you need to plug in at least one point on that line now luckily we do have the coordinates of K and so we can plug in the zero and the minus two and a quarter into that equation so there's so the two and a quarter the minus two and a quarter is the Y value so I'm going to plug that over here so minus two and a quarter by the way I don't like to work with mixed numbers so I'm going to convert that into an improper fraction so that'll be nine over four so that'll become nine over four equals to a half and then X is zero and then plus C so if you had to go work out C you're going to get minus 9 over 4. okay so then the final answer for that question uh let's write it down here y equals to a half X minus nine over four okay y equals to a half x minus nine over four and the last one write down the coordinates of C okay now this is a nice easy one the reason is is that this shape here is a parallelogram right this um a b c d yes it's also a rectangle but because it's a parallelogram there is a very easy way that we can work out this fourth coordinate what you do is you look at this point here and then you look at this point here now how do you go from a to d so to go from a to d what do we do to the X values well um U minus four see that you go from zero to minus four so you minus four on The X and then what do you do to the Y values U minus 2 because that is how you go from four to two so you minus 2. so for the X values we minus four and for the Y values we minus two so we're going to do the exact same thing to go from B to C so to go from B to C we're also going to take the X values and we're going to minus 4 and we're going to take the Y values and we're going to minus 2. and so if you're to minus 4 from that x value you're going to end up with zero and if you had to minus 2 from that y value you're going to end up with negative 6. and so the answer will be 0 and negative 6 and the reason is just called inspection the other way that you could have done that is you could have realized that this is the diagonal okay and earlier on we worked out the length of diagonal BD can you remember it was one of the first questions we did in this question and we got an answer of uh 10. now what we should know is that the diagonals of a rectangle they are equal in length so what we can do is we can say that um the diagonal or we can actually just say sorry we can say that AC the length of AC is equal to the length of BD and that's just because of diagonal of rectangle okay and so that means that the length of AC is 10. so if you just go 10 units down from 0 and 4 then what would you end up with well if you take the zero will stay a zero um because you on the X you I mean you're on the y-axis so that's not going to change but the four if you take away 10 that becomes minus 6 and so you would end up with 0 and -6 but that's exactly what we got using the inspection method earlier