Question

what are the coordinates of the circumcenter of \\( \triangle abc \\)?
enter your answer in the boxes.
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Explanation:

Step1: Recall the property of circum - center for a right - triangle

For a right - triangle, the circum - center is the mid - point of the hypotenuse.

Step2: Identify the vertices of the right - triangle

From the graph, \(A(-5,-4)\), \(B(3,-4)\), \(C(-1,6)\). Since \(AB\) is horizontal (\(y = - 4\) for both \(A\) and \(B\)) and \(AC\) and \(BC\) are non - horizontal and non - vertical, \(\triangle ABC\) is a right - triangle with hypotenuse \(AB\) (using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), slope of \(AB\) is \(m_{AB}=\frac{-4+4}{3 + 5}=0\), slope of \(AC\) is \(m_{AC}=\frac{6 + 4}{-1+5}=\frac{10}{4}=\frac{5}{2}\), slope of \(BC\) is \(m_{BC}=\frac{6 + 4}{-1 - 3}=-\frac{5}{2}\), and \(m_{AC}\times m_{BC}=-1\) (if it was a non - right triangle, we would find the perpendicular bisectors of two sides).

Step3: Use the mid - point formula

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Here \(x_1=-5,y_1=-4,x_2 = 3,y_2=-4\).

$$x=\frac{-5 + 3}{2}=\frac{-2}{2}=-1$$

$$y=\frac{-4-4}{2}=\frac{-8}{2}=-4$$

Answer:

\((-1,-4)\)