Question

Find the coordinates of the circumcenter of DEF.

11. D (6,0), E (0,6), F (-6,0)
12. D (0,0) E (6,0), F (0,4)

Answer 1

1. The coordinates of the circumcenter of triangle DEF with the given vertices are (0, 0).
2. The coordinates of the circumcenter of triangle DEF with the given vertices are (-1/3, 3).

How to determine the coordinates?

In order to determine the coordinates of the circumcenter of triangle DEF with the given vertices, let its circumcenter be P(x, y). Hence, PD = PE = PF.

By squaring all sides of triangle DEF, we have:

PD² = PE² = PF²

From PD² = PE², we have:

(x - 6)² + (y - 0)² = (x - 0)² + (y - 6)²

x² - 12x + 36 + y² = x² + y² - 12y + 36

-12x + 12y = 0    ..........equation 1.

From PE² = PF², we have:

(x - 0)² + (y - 6)² = (x - (-6))² + (y - 0)²

x² + y² - 12y + 36 = (x + 6)² + y²

x² + y² - 12y + 36 = x² + 12x + 36 + y²

-12x - 12y = 0     ..........equation 2.

Solving eqn. 1 and eqn. 2 simultaneously, we have:

x = 0 and y = 0.

Therefore, the coordinates of the circumcenter of triangle DEF with the given vertices are (0, 0).

At D (0,0) E (6,0), F (0,4).

From PD² = PE², we have:

(x - 0)² + (y - 0)² = (x - 6)² + (y - 0)²

x² + y² = x² - 12y + 36 + y²

12y = 36    

y = 3.

From PE² = PF², we have:

(x - 6)² + (y - 0)² = (x - 0)² + (y - 4)²

x² - 12x + 36 + y² = x² + y² - 8y + 16

-12x - 8y = -20    

-12x - 8(3) = -20

-12x - 24 = -20

12x = -4

x = -1/3.

Therefore, the coordinates of the circumcenter of triangle DEF with the given vertices are (-1/3, 3).

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