Geometry 9 1 Polygons in the Coordinate Plane

hello class welcome to geometry lesson 9 1 which is all about polygons in the coordinate plane by the end of this lesson you should be able to use the coordinate plane to analyze geometric figures so this is going to combine bunch of previous knowledge in this chapter ok so if we look at this first example we want to say connect algebra and geometry through coordinates so what formulas can be used to identify properties of figures on the coordinate plane so letter which formula can use to find so if we're thinking about aids right here is right here and if want to find the distance between them I'm going to use the distance formula so for those of you that don't really remember how to do the distance formula quick refresher it's when you take your x-values and find the difference between them you square it and then you find the difference between your y-values and you square it and you add those two numbers together so in this case would be taking the square root of let's see an x-value have let's see have 3 so could say 3 minus my other value is 1 squared and if you wanted to do 1 minus 3 that's okay and then since started with the 3 need to start with the value from that same point so that I'm gonna say 1 minus 6 and if you for the first parenthesis said 1 minus 3 your second parentheses would say 6 minus 1 okay so 3 minus 1 is 2 2 squared is 4 1 minus 6 is negative 5 negative 5 squared is 25 that means the distance between them is the square root of 29 ok letter what point bisects segment a-b remember bisect means cut in half or half way so what's halfway between and that means we're going to use midpoint formula so in order to find our midpoint I'm going to add my X's together so have 3 and 1/2 my y's 1 & 6 so I'm going to say 1 plus 6 divided by 2 so 3 plus 1 gives me 4 4 divided by 2 is 2 1 plus 6 is 7 7 divided by 2 I'm just going to leave as fraction and that is the point that bisects ok so distance formula midpoint formula letter why do slopes of segment and segment BC show that the measure of angle ABC equals 90 degrees so slope remember it's been while slope is y2 minus y1 over x2 minus x1 and if they are 90 degrees that means our slopes are opposite reciprocals if you remember so if have 1/2 my perpendicular slope is negative 2 over 1 so that's the situation we're looking for let's find the slope of segment so I'm going to take my y-values so if start with 6 and then my other y-value I'm gonna say -1 since started up here at the 6 for the that means could start with the 1 for so I'm gonna say 1 minus 3 so 6 minus 1 is 5 1 minus 3 is negative 2 that's the slope for now the slope for BC let's see if take the value have 3 minus 1 over 8 minus 3 so 3 minus 1 is 2 8 minus 3 is 5 so are these sit reciprocals so remember reciprocal means that you take fraction and you flip it so in this case started with five on top and two on bottom and then for the next one had two on top and five on bottom so they're reciprocals and the opposite just means one's positive ones negative in this case they are so the reason they show that the that it's right angle is because they have opposite reciprocal slopes so there's all of your previous knowledge distance formula midpoint formula and slope now we're gonna use that repeatedly throughout this lesson but why don't you look at this triangle ABC and find what is the length of the line line segments are connecting the midpoints of segments and Sigma BC so want the length of line so that tells me I'm gonna do distance but I'm gonna do distance with the midpoint so first start by finding the midpoints then find the distance between the midpoints good luck hopefully for your answer you ended up with the square root of seven point two five but if you did not get the square root of seven point two five and you said instead two point six nine that is also correct answer okay now let's talk about classifying triangle on the coordinate plane so if want to figure out if something's equilateral isosceles are stealing that means I'm going to be doing the distance formula on on each side of the figure in order to figure out if they have the same measures or different measures remember equilateral triangles have three congruent sides isosceles have two congruent sides and scaling have no congruent sides okay so I've already refreshed your memory on distance formula so decided not to write it out again the book did it for us XY they figured out they found that it's the square root of 52 square root of 13 and square root of 65 since there all different the triangle is scalene and then letter if want to figure out if it's right triangle mean you could go through and you can find all of the slopes and see if you have any opposite reciprocals but faster way if you already know what the three measures are we have Pythagorean theorem can say the square root of 52 squared is let's quickly refresh our memories on that remember squares and square roots cancel each other so then you're left with just whatever's under the square root so in this case that's 52 so then this ends up becoming 52 plus 13 does that equal 65 it does so we know that this is right triangle and make sure that when you set up Pythagorean theorem to try to figure out if it is right triangle the side that's the biggest always needs to be on its own why don't you go ahead and solve this next one good luck all right for this problem hopefully you ended up with two sides that were the same length so you knew that they were or that this forms an isosceles triangle and you figured out if you do Clarion Cara it doesn't work because our two congruent sides happen to also be the longest sides so we would say that they are not they don't form right triangle another quick side note this is another chapter where we're going to practice simplifying radicals so the square root of 40 could split up into the square root of 4 times the square root of 10 which becomes 2 square root of 10 just like 32 could split up into the square root of 4 times the square root of actually the square root of 16 times the square root of 2 which becomes 4 square root of 2 okay so you will have to practice that this chapter all right example 3 let's talk about classifying parallelogram or no plate so if want to figure out what type of parallelogram is rst you so remember parallelograms are kind of the biggest category of quadrilaterals that have two opposite sides or two sets of opposite sides that are parallel parallelograms that are more specifically it becomes rectangle and rhombus and then if it's all of those it is square so we're gonna go through the process to see what figure do we have okay so what type of parallelogram is rst you so first we have to find the distances and you don't have to do it on all four sides because if we know it's parallelogram we know we have two sets of congruent sides so that means would only have to do it on let's see our RS because that's the same as UT or are sorry and would have to do it on either st or are you because those are the same so if do that distance formula get square root of 52 and square root of 10 and then since not all side lengths are equal RS to you is not rhombus or square because remember rhombi and squares have all four sides are congruent so if you got the same measure for both sides immediately you're going to think maybe it's rhombus or square so now we know that it's not around it's not square so that leaves us with parallelogram or rectangle so in order to know if it's rectangle what can test is the slopes of the sides to see five right angles so if find the slope of st and compare it to one of the shorter sides if it's right if it's rectangle there'll be opposite reciprocal slopes so if find slope of st end up with negative 2/3 if find the slope of RS end up with three those are not opposite reciprocals therefore they do not form 90-degree angle so can rule out rectangle and that means that this is just parallelogram and not rectangle rhombus or square why don't you go ahead and take look at this problem on your own good luck all right hopefully for this question you figured out it's not rhombus because we do not have congruent sides but it is rectangle because if we check the slopes we have opposite reciprocals so they form right angles all right one last thing on that question recommend if you're ever just given vertices maybe try just even rough sketch of plotting them out so you have nice visual so you end up choosing the correct sides sometimes see students that without the visual they end up trying to compare to DC and then they're like cool have congruent sides that means that this is rectangle or this is rhombus or square but that they got false information because they chose opposite sides which are always going to be congruent in parallelogram okay so example 4 what if we want to check for trapezoid or kite so you're still using the same tests remember trapezoid has exactly one pair of parallel sides so we've talked about perpendicular slope parallel slope is when they're exactly the same okay so that means to check for trapezoid you do the slope of all four sides so they did found the slope of was negative 2/3 BC was 1/2 was 4 and AD was 1/2 so if you notice we have two sides that are parallel to each other and since we have one set of parallel sides that means that this is trapezoid that is all you have to do to check for trapezoid ok and then if we want to know if something's like Heights the way you do that is kite has two pairs of consecutive congruent sides and no opposite sides so for kite you're going to use the distance formula to find the distance of all four sites and you can see right away okay have have two sets of congruent sides but the key part of making this kite is they have to be right next to each other so again this is we're trying rough sketch of where those points are really helps so KL needs to be congruent to LM and it is they're both the square root of 10 and the same thing goes for and pulse need to be congruent as well and since they are that means that we have kite so think you guys have seen the patterns right now you're doing lot of distance formula slope you're doing lot of midpoints it's lot of the same thing and then just knowing what makes something parallelogram rectangle rhombus square kite trapezoid all of those things so want you to check to see it is either quadrilateral kite trapezoid or neither all right hopefully for this problem you said that letter is kite because you have two sets of consecutive sides that are congruent and then letter is actually neither so if you find the slope none of the slopes are parallel and also if you look at the sides you can see that we don't really have two sets of congruent sides going on either so totally just from visual standpoint skipped the kite test for that one okay example five so I'm not going to go through and actually solve this just want to talk you through how you would solve it so if we look at this example we're talking about finding perimeter and area so Dylan draw some plan to fence in yard for his chickens the distance between grid lines is 1 foot so we want to know is 30 feet of fencing enough to enclose the yard the way that we would figure that problem out is we'd find the distance of all three sides and we would add them together and we would see if take the lengths of all three sides added together is it less than if it is have enough fencing if it's not less than 30 then need more fencing or Dylan needs more fancy guess the more complicated problem in this example is finding the area so to find the area of triangle you guys know is 1/2 base times height you guys have been working with that for really long time by now and the really important thing to know about finding the area on this triangle is if well first of all if had already done letter even know that is 10 would know that BC is 10 and then this other side would have found is the square root of 80 okay so that's what would have found in letter letter now what would use is need to find the height of this triangle remember height of triangle always has to come from right angle it has to come from an altitude so we have to find what that distance is so that means have to find the midpoint of AC so if found the midpoint of AC would say 2 plus 10 is 12 divided by 2 and then would say 6 plus 2 is 8 divided by 2 so my midpoint is at 6 4 and then what would do is would find the distance between my midpoint right here end of this vertice right here and that would give me my height and if were to do that would find that the height of the triangle is also square root of 80 it does not always work out that matches the bottom but in this case it does so then would do one-half times the base which is square root of 80 times the height which is square root of 80 and that gives me my area so the square root of 80 times square root of 80 just becomes 80 and I'm multiplying that by one-half and get 40 yards squared okay so that is how you would solve that problem to find the area first find the altitude which you do by finding the midpoint of the side and the distance between the midpoint and the opposite vertices and then you can plug it into the area of triangle I'm gonna have you skip that and here's our concept summary if we look at this concept summary it helps you figure out all of the different techniques you will use to determine what type of shapes things are recommend starting chart of what tests you need to do in order to determine what figure is all right if you have any questions at all please feel free to reach out for spell and I'm happy to help you have great day