Question

find the coordinates of the circumcenter of DEF with coordinates D(1,3) E(8,3) and F(1,-5). Show your work

Answer 1

As shown in attached figure C is the circumcenter of triangle DEF

First find midpoint of DF using mid point formula

$(\frac{(x_{1} + x_{2})}{2} , \frac{(y_{1} + y_{2})}{2})$----------------------------(1)

for two points $(x_{1} ,y_{1} )$ and $(x_{2} ,y_{2} )$

so mid point M of D(1,3) and F(1,-5) will be

$(\frac{(1+1)}{2} , \frac{(3-5)}{2})$

$(\frac{2}{2} , \frac{-2}{2})$

M : (1,-1)

Find slope of DF using slope formula

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$-----------------------------------------(2)

So slope of D(1,3) and F(1,-5) will be will be

$\frac{-5-3}{1-1}$

$\frac{-8}{0}$

Now line CM is the perpendicular bisector of line DF. So slope of line CM will be negative reciprocal of slope of DF

So slope of CM will be $\frac{0}{8}$ or simply 0

So now we will write equation of line CM using slope m =0 and passing through mid point M (1,-1) using formula

$y - y_{1} = m(x - x_{1} )$

on plugging values we get

$y - (-1) = 0(x - 1)$

y + 1 =0

y +1 -1 = 0 -1

y = -1 ---------------------------------------------------(3)

So thats the equation of line CM

Similarly now line midpoint N of line DE using mid point formula from (1)

so mid point N of D(1,3) and E(8,3) will be

$(\frac{(1+8)}{2} , \frac{(3+3)}{2})$

$(\frac{9}{2} , \frac{6}{2})$

N : (4.5,3)

Find slope of DE using slope formula in (2)

So slope of D(1,3)) and F(8,3) will be will be

$\frac{3-3}{8-1}$

$\frac{0}{7}$ or simply 0

Now line CN is the perpendicular bisector of line DE. So slope of line CN will be negative reciprocal of slope of DE

So slope of CN will be be reciprocal of 0 which is undefined when 0 goes in denominator. All undefined slope lines have equation of form $x = x_{1}$ for point $(x_{1} ,y_{1} )$ on it

So for line CN with undefined slope and passing through point (4.5,3) its equation will be

x = 4.5 ----------------------------------------------------------------------------(4)

Solving equations (3) and (4) we get coordinates of intersection point C which will be

x=4.5, y =-1

so answer for circumcenter C coordinates will be (4.5,-1)