Find the coordinates of the circumcenter for ∆DEF with coordinates D(1,1) E (7,1) and F(1,5).
can you please explain how i can find this answer? i am having trouble
the circumcenter is equidistant from all three vertices
it lies on the intersection of the three perpendicular bisectors of the sides
midpoint DF is ... (1,3) ... b DF is ... y = 3
midpoint DE is ... (4,1) ... b DF is ... x = 4
this might be a good place to start:
https://brilliant.org/wiki/triangles-circumcenter/
You can also work out where the perpendicular bisectors of two of the sides meet.
steve
i got -2a + 1 - 2b+1 = 14a+28 + 2b+1 i dont know what to do after
and
14a + 28 - 8b-16 + 2a + 1 - 10b+20 but i dont know what to do after this one too :/
what, you forgot your Algebra I ?
-2a + 1 - 2b+1 = 14a+28 + 2b+1
14a + 28 - 8b-16 = 2a + 1 - 10b+20
kinda messy, so simplify a bit
16a+4b = -27
12a+2b = 9
Now you can use those to solve for a and b, which I assume are the coordinates of the circumcenter.
are we all solving the same problem?
I didnt forget i just have trouble understanding as quick and easy as others im not good in math...
but thankyou for helping me.
Certainly! To find the coordinates of the circumcenter for triangle ∆DEF, you can follow these steps:
1. Calculate the midpoints of two sides of the triangle.
- The midpoint between points D and E can be found by averaging their x-coordinates and y-coordinates separately. So, the midpoint of DE is ((1+7)/2, (1+1)/2) = (4,1).
- The midpoint between points E and F can be found similarly. The midpoint of EF is ((7+1)/2, (1+5)/2) = (4,3).
2. Find the slopes of the perpendicular bisectors of two sides of the triangle.
- The slope of the perpendicular bisector of DE can be calculated by taking the negative reciprocal of the slope of DE. The slope of DE is (1-1)/(7-1) = 0/6 = 0. Therefore, the slope of the perpendicular bisector is undefined (since the negative reciprocal of 0 is undefined).
- Similarly, the slope of the perpendicular bisector of EF can be calculated. The slope of EF is (5-1)/(1-7) = 4/-6 = -2/3. Thus, the slope of the perpendicular bisector is 3/2 (the negative reciprocal of -2/3).
3. Find the equations of the perpendicular bisectors using the slopes and midpoints.
- The equation of the perpendicular bisector of DE can be written as y - y1 = m(x - x1), where (x1, y1) is the midpoint (4,1) and m is the slope (undefined). Therefore, the equation becomes x = 4.
- Similarly, the equation of the perpendicular bisector of EF can be written as y - y1 = m(x - x1), where (x1, y1) is the midpoint (4,3) and m is the slope (3/2). Thus, the equation can be expressed as y = (3/2)x - 3.
4. Solve the simultaneous equations of the perpendicular bisectors.
- By solving the simultaneous equations x = 4 and y = (3/2)x - 3, you can find the point where the two perpendicular bisectors intersect, which is the circumcenter.
- Substituting x = 4 into the equation y = (3/2)x - 3, you get y = (3/2)(4) - 3 = 6 - 3 = 3. So, the coordinates of the circumcenter are (4,3).
Therefore, the circumcenter of triangle ∆DEF is at coordinates (4,3).