Question

How do you find the circumcenter of a triangle?

Answer 1

Find the circumcenter of \[\Delta EFG \] with \[E(2,6), F(2,6), G(6,4)\]

Answer 2

do you know what the circumcenter of a triangle is?

Answer 3

isnt it where all of the medians intersect

Answer 4

no, that's the centroid...

Answer 5

hey i was writing that down

Answer 6

the point where all the perpendicular bisectors of the sides meet is called the circumcenter...

Answer 7

sorry... i wanted to get the terms right....

Answer 8

how do i find it do i graph the triangle or..? and thats ok

Answer 9

no need to graph.... just use algebra...
if we can find the equation of two perpendicular bisectors and find where they intersect, then we're done...

u wanna do it the algebra way?

Answer 10

yes how do we set that up o:

Answer 11

graphing MIGHT be easier though...

Answer 12

i should probably know the algebra way for tests when theres no graph paper..

Answer 13

ok... first we need to find the midpoints of two sides.... let's pick EF and FG.
so find the midpoint of E(2, 6), F(2, 6)
wait E and F have the same coordinates?

Answer 14

Yep evidently

Answer 15

wait no

Answer 16

E(2,6) and F(2, 4)

Answer 17

i typoed

Answer 18

ok find the midpoint of E(2,6) and F(2,4) using the midpoint formula...
do you remember the midpoint formula?

Answer 19

what was it again

Answer 20

it was like x2/x1 or something

Answer 21

midpoint = (average of the x's, average of the y's)

Answer 22

mipoint = \(\large (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \)

Answer 23

\[(\frac{ 2 + 2 }{ 2 }\ ), (\frac{ 6 + 4 }{ 2 })\]

Answer 24

ok... simplified you have the midpoint of EF as (2, 5) correct?

Answer 25

umm how did u get 5

Answer 26

6+4=10
10/2=5

Answer 27

oh -_- sorry ok so what do we do next

Answer 28

5?

Answer 29

no 2

Answer 30

what is the slope that is PERPENDICULAR to side EF?

Answer 31

wait wht hold on

Answer 32

2?

Answer 33

slope = \(\large \frac {y_2-y_1}{x_2-x_1} \)

Answer 34

\[\frac{ 4-6}{ 2-2 } =\frac{ -2 }{ 0 }\]

Answer 35

slope is -2

Answer 36

ok... you have a denominator of zero, which means the slope of EF is undefined...
so the slope that is perpendicular to EF has slope 0, agreed?

Answer 37

right

Answer 38

so the perpendicular bisector of EF has slope 0, so the equation of the line through (2,5) is y=5...
agreed?

Answer 39

actually there is an alternate definition of circumcenter which will help us solve the question quicker..
circumcenter is the point which is equidistant from all the 3 vertices of the triangle..

Answer 40

if you plot the points, you'll be looking at the vertices of a right triangle.

Answer 41

:'s

Answer 42

which is obviously evident from any figure..

Answer 43

this is a right triangle?

Answer 44

well i wanted to know the algebraic way so that i can do it when i dont have graph paper

Answer 45

It is?

Answer 46

you may continue with what you've started. but as a precautionary, check first if the triangle is a right triangle because if it's a right triangle, the solution is very simple.

Answer 47

yes it is..
and yes, i shuda realized that too....
@shubhamsrg is correct then, the easier way would be to take the mipoint of the hypotenuse.

Answer 48

well how do we do that

Answer 49

exactly!

Answer 50

uh.... like you said... maybe we can graph it and see which is the hypotenuse....
(don't kill me!!!)

Answer 51

:'s..

Answer 52

i ripped my notes paper erasing -__- which are we actually gonna do lol

Answer 53

don't do that.. the procedure we went through works on ALL triangles....

Answer 54

..but we only did 2 steps

Answer 55

that's ok.... because the other steps is doing the same thing to another side...

Answer 56

anyways, if you dont know what kinda triangle it is,, just use distance formulla twice, you'll get 2 linear eqns which are easier to solve..

Answer 57

well can we hurry its only a few hours till morning here and ive got to get this assignment done.. and i have no idea how to do this

Answer 58

@shubhamsrg , be my guest....???? pls... i need to be quiet now.... sorry @icalibear .... :(

Answer 59

D':

Answer 60

EF is vertical, FG is horizontal. so EG is the hypotenuse.

Answer 61

ok got it

Answer 62

wats next 0.0

Answer 63

i also have to leave now..sorry! :P
anyways, i can help a bit..

let O be circumcenter
we know
OA = OB =OC

you must be knowing distance formula, use that in any 2 equalities and solve for x and y

also, you can find out that this is a right triangle which will then be very easy to solve..