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To find the circumcenter, find the intersection of two perpendicular bisectors of two sides of the triangle.
(7,5)
Let's start by graphing the triangle using the given coordinates.
To find the circumcenter, we need equations for the perpendicular bisectors of at least two sides of the triangle. Recall that a bisector cuts something in half, so we want to find lines that are perpendicular to the sides at their midpoints.
Let's first find the midpoints of two sides. To do so, we can use the Midpoint Formula.
| Side | Points | M(x_1+x_2/2,y_1+y_2/2) | Midpoint |
|---|---|---|---|
| DF | ( 3,5), ( 11,5) | G(3+ 11/2,5+ 5/2) | G(7,5) |
| EF | ( 7,9), ( 11,5) | H(7+ 11/2,9+ 5/2) | H(9,7) |
According to the Slopes of Perpendicular Lines Theorem, the product of the slopes of perpendicular lines is -1. This allows us to find the slope of the perpendicular bisector from the slope of the side. Since we know the endpoints of EF, we can find its slope using the Slope Formula. m = y_2-y_1/x_2-x_1 Let's substitute the given endpoints E(7,9) and F(11,5).
We found that the slope of EF is -1. Now we can find the slope of the perpendicular bisector from the product. Let's call it m_p. -1* m_p = -1 ⇒ m_p = 1 The slope of the perpendicular bisector of EF is 1. y=1x+b ⇔ y=x+b To complete its equation, we need the y-intercept. We can use the fact that the line passes through the midpoint H(9,7).
Therefore, the equation of the perpendicular bisector of EF is y=x-2. To find the second one, notice that DF is horizontal. This means that the perpendicular bisector of DF is a vertical line. Moreover, since it passes through G( 7,5), its equation is x= 7.
We want to find the intersection of the two perpendicular bisectors that we found. To do it, we need to solve the system of their equations. x=7 y=x-2 We can solve it by substituting x=7 in the second equation.
Therefore, the coordinates of the circumcenter are (7,5). It is the midpoint G.
