Angle bisectors
Inradius Perimeter and Area Showing that area is equal to inradius times semiperimeter
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- We are told that triangle ABC has perimeter p and inradius r
- and then they want us to find the area of ABC in terms of p and r
- so we know that perimeters are just the sum of the sides of the triangle
- or how long of length will have to be if you wanna to go around the triangle
- and let's just remind ourselves what the inradius is
- if we take the angle bisectors of each of these vertexes
- each of these angles right over here
- so bisectors that right over there
- and then bisector that right over there
- this angle is going to be equal to that angle
- this angle is going to be equal to that angle
- and then this angle is going to be equal to that angle there
- and the point where those angle bisectors intersect
- that right over there is our incenter
- and it is equal distant from all of the three sides
- and the distance from those sides that's the inradius
- so let me draw the inradius
- so when you find the distance between the point and the line
- you wanna drop a perpendicular
- so this length right over here is the inradius
- this length right over here is the inradius
- and this length right over here is the inradius
- and if you want you could draw a incircle here with the center at the incenter
- and with the radius r and that circle would look something like this
- we don't have to actually draw for this problem
- so you could draw a circle that looks something like that
- then we call that the incircle
- so let's think about how we can find the area here
- especially in terms of this inradius
- well the cool thing about the inradius is this it looks like the altitude
- well this looks like the altitude for this triangle right over here
- triangle A let's label the center let's call it I for incenter
- so r is this r right over here is the altitude of triangle AIC
- this r is the altitude of triangle BIC
- and this r which we didn't label
- that r right over there is the altitude of triangle AIB
- and so we could find the area of each of those triangles in terms of both r and their basis
- maybe if we sum up the area of all the triangles
- we can get something in terms of our perimeter and our inradius
- so let's just try to do this
- so the area of the entire triangle the area of ABC is going to be equal to
- and I will color code this
- this is going to be equal to the area of AIC
- so that's what I am shading here in magenta is going to be euqal to the area of AIC
- plus the area of BIC which is this triangle right over here
- I will show you that in a different color I have already used blue
- so let me do that in orange
- plus the area of BIC so that's this area right over here
- so plus the area of BIC and then finally plus the area
- I will do this in let's see I will use this pink color
- plus the area of AIB that is the area AIB
- take the sum of the areas of these three triangles
- you've got the area of the larger triangle
- Now AIC the area of AIC is going to be equal to one half base times height
- so this is going to be one half the basis of the length of AC one half AC
- times the height times this altitude right over here
- which is just going to be r times r
- that's the area of AIC
- and then the area of BIC is going to be one half the base which is BC
- times a height which is r
- and then plus the area of AIB this one right over here is going to be
- one half the base which is the length of the side AB
- AB times a height which is oneeagain r
- and over here we can fetch out one half r for all of these terms
- and you get one half r times AC plus BC plus AB
- and I think you see where this is going
- plus that's the different shade of pink plus AB
- Now what is AC plus BC plus AB
- well that's going to be the perimeter p if you just take the sum of the sides
- that is the perimeter of p and it looks like we are done
- the area of our triangle of ABC is equal to one half times r times the perimeter
- which is kind of a neat result
- one half times the inradius times the perimeter of the triangle
- or sometimes we will see it written like this is equal to r times p over s
- oh sorry p over 2
- and this term right over here the perimeter divided by 2
- is sometimes called semi perimeter and sometimes it is denoted by s
- so sometimes you will see the area is equal to r times s
- where s is the semi perimeter it's the perimeter divided by 2
- I firstly like this way a little bit more
- because I remember that p is the perimeter
- this is useful because obviously now if someone gives you inradius and a perimeter
- you can figure out the area of a triangle
- or someone gives you the area of the triangle and the perimeter
- you can get the inradius of it
- if given any two of these variables you can always get the third
- so for example if someone if this was a triangle right over here
- which is the most famous of the right triangles
- if I have a triangle that has length 3 4 and 5
- we know this is a right triangle
- you can verify this from the pythagorean theorem
- and someone says what is the inradius of this triangle right over here
- well we can figure out the area pretty easily
- we know that this is a right triangle 3 squared plus 4 squared is equal to 5 squared
- so the area is going to be equal to 3 times 4 times one half
- so 3 times 4 times one half is 6
- and the perimeter here is going to be equal to 3 plus 4 which is 7 plus 5 is 12
- and so we have the area so let's write this
- the area is equal to one half times the inradius times the perimeter
- so here we have 12 is equal to one half times the inradius times the perimeter
- so we have oh sorry we have 6 let me write this
- the area 6 6 is equal to one half times the inradius times 12
- and so in this situation one half times 12 is 6
- 6 is equal to 6r divide both sides by 6 you get r is equal to 1
- so if you want to draw the inradius for this one which is kind of a neat result
- so let me draw some angle bisectors here
- this 3-4-5 right triangle has inradius of 1
- so this distance equals this distance which is equal to this distance
- which is equal to 1 just kind of a neat result
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