- The coordinates of the circumcenter of triangle DEF with the given vertices are (0, 0).
- The coordinates of the circumcenter of triangle DEF with the given vertices are (-1/3, 3).
<h3>How to determine the coordinates?</h3>
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).
<h3>At D (0,0) E (6,0), F (0,4).</h3>
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).
Read more on circumcenter here: brainly.com/question/28027775
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