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## Class 10 (Old)

### Course: Class 10 (Old) > Unit 7

Lesson 4: Area of a triangle# Area of triangle from coordinates example

Let's find the area of a triangle when the coordinates of the vertices are given to us. Let's do this without having to rely on the formula directly. Created by Aanand Srinivas.

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- And I thought my approach was smart! (Poor me). I found the length of a side, then found the equation of the perpendicular line passing through the opposite vertex of the side, did a system to find the intersection of the line of the base with that of the height, calculated the distance between the base and the opposite vertex and then used the triangle's formula....

I have no words :')(6 votes)

## Video transcript

find the area of the triangle whose vertices are -5 comma 1 3 comma -5 and 5 comma 2 the if you start thinking about it I first want the diagram a picture in my mind so I'm gonna have my corner axis and what's my first point -5 comma 1 - 1 - 2 - 3 - 4 - 5 comma 1 3 comma - 5 3 comma minus 5 the second point over here and 5 come out to the third points here so my triangle should look something like that let's draw it so -5 comma 1 3 comma -5 and 5 comma 2 now the problem why this is even a difficult problem is that these lines are not vertical or horizontal if they had been I try to find the base and height and just multiply but to find this area now I need to do something clever because it's rectangular coordinates which is just basically our usual coordinate geometry with the x-axis and y-axis I know that all these lengths are in terms of horizontal distances or vertical distances so if I can somehow cleverly use that I can find the area much easily so what I'm gonna do is I'm gonna draw some horizontal lines over here let's draw it one over here vertical lines like this vertical lines over here oh sorry horizontal lines over here and vertical lines over here now notice that now that I've done this I have some shapes that I can use to find the area of my triangle which is I need to find the area of this big rectangle subtract this triangle this triangle this triangle why that is easier than finding the area of this triangle is that all of them have horizontal sides and vertical sides and that's it so let's jump in and do this and if you feel like it you should pause the video now and see if we can do this by yourself because after this it's about looking at all these lengths finding the areas and subtracting to get the area of this triangle I'm gonna do it now so I'm going to first find the rectangle area that's the area that I need to first find so that I can subtract the areas of these three triangles so rectangle area let's call it rectangle area and what is that going to be equal to length into breadth but it's the length and web given to me it is just not very directly if you notice then this length over here has to be the difference in the x-coordinates between this point and this point because this point and this point have the same x-coordinate so what is that going to be the x-coordinate here is five the x-coordinate here is minus five so the length here is ten units we don't know what whether it's centimeters or millimeters or kilometers but we just know it some 10 units and I need the Brit now and what is that going to be equal to the height of the breadth how you want to think about it is going to be equal to the difference in the y-coordinates between this point which is the same as over here and this point which is the same as this line so the y-coordinate here is minus five the y-coordinate here is 2 so minus Phi all the way to 2 this length will be 7 so the area of the rectangle is 7 into 10 or 70 square units units that's part one now I need to find the area of these three triangles let me do it one by one let me try to shade that so let me shade this now here this is a triangle that I want but to find the area of this triangle I need the base and the height and then I can do half into base into height so what is the base what is this going to be that's going to be equal to the difference in the x coordinate between this point and this point that's 3 minus minus 5 that's all the way from 3 till minus 5 it's 8 that's it this length is 8 and what is this height going to be you can start noticing how you're doing the same thing for each of these so the height is 1 the y coordinate here is 1 the y-coordinate here is minus 5 so this will go to 0 and then all the way till minus 5 so 1 plus 5 and 6 so this height is 6 this base is 8 so this area over here is going to be equal to 8 into 4 because 18 oh sorry 6 into 4 right 6 into 8 and 1/2 1/2 into 6 into 8 which is 24 that's the area of this triangle what is the area of this triangle you can pause right now and do it and check if we're doing it similarly so the area of this triangle F I need the base for it and the height the base is 5 minus 3 the difference in the x-coordinates and that's 2 and the height is the difference in their y-coordinates which is 2 minus minus 5 all the way to here and then another five years of 7 so 7 into 2 into 1/2 or another was just 7 so let's shade this and then we write it 7 so that's the area of this triangle we finally need the area of this triangle this little one over here so that's going to be equal to the base into the height into 1/2 the base here is the difference in the y-coordinates 2 + 1 which is just 1 so that's why it looks small our diagram is not bad and the difference in the y coordinate is sorry the x coordinate is 10 we already found it in fact I just noticed that we didn't have to do this 7 again we already did that to find the rectangle I could have just used it I wasted some of my own time so here I have 10 over here so this is also going to be 10 so 1/2 into 10 into 1 so that's 5 so let's write that maybe in yellow this one here is 5 so now to find the area of my area of triangle what I have to do is take my area of the rectangle which is 70 and I subtract the 24 and I subtract my 7 and I subtract 5 and whatever I get is my area of the triangle so 70 minus 20 is 50 minus 4 is 46 46 that's right minus seven is 39 minus 5 is 34 34 square centimeters so that's the area of this triangle so it's marker over here 34 which is what we wanted square centimeters know that square units know units have been given to us so it's square units and what we've actually done here is the formula you may have seen that X 1 into y 2 minus y 1 plus so that formula we've actually done the same thing here except that we've kind of read arrived it so even if you hadn't used the formula you'd have got the same answer