Question

D(−1, 8), E(4, −2), F(−5, −3) are midpoints of sides BC, CA and AB of ∆ABC Find co-ordinates of the circumcenter of ΔABC

Answer 1

Here, A(0, −13), B(−10, 7), C(8, 9) are the vertices of ΔABC.

Let F be the circumcentre of ΔABC.

Let FD and FE be perpendicular bisectors of the sides BC and AC respectively.

∴ D and E are the midpoints of side BC and AC.

∴ D ≡ `((-10 + 8)/2, (7 + 9)/2)`

∴ D = (−1, 8)

and E ≡ `((0 + 8)/2, (-13 + 9)/2)`

∴ E = (4, −2)

Now, slope of BC = `(7 - 9)/(-10 - 8) = 1/9`

∴ Slope of FD = − 9      ...[∵ FD ⊥ BC]

Since FD passes through (−1, 8) and has slope −9, equation of FD is

y − 8 = −9(x + 1)

∴ y − 8 = −9x − 9

∴ y = −9x − 1    ...(i)

Also, slope of AC = `(-13 - 9)/(0 - 8) = 11/4`

∴ Slope of FE = `(-4)/11`  ...[∵ FE ⊥ AC]

Since FE passes through (4, −2) and has slope `(-4)/11`, equation of FE is

y + 2 = `(-4)/11(x - 4)`

∴ 11(y + 2) = −4(x − 4)

∴ 11y + 22 = −4x + 16

∴ 4x + 11y = − 6     ...(ii)

To find coordinates of circumcentre, we have to solve equations (i) and (ii).

Substituting the value of y in (ii), we get

4x + 11(−9x − 1) = − 6

∴ 4x − 99x − 11 = − 6

∴ −95x = 5

∴ x = `(-1)/19`

Substituting the value of x in (i), we get

y = `-9((-1)/19) - 1 = (-10)/19`

∴ Co-ordinates of circumcentre F ≡ `((-1)/19, (-10)/19)`.