If a circle passes through points (1, 2), (2, 5), and (5, 4), what is the diameter of the circle?

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We need to find the diameter of a circle that passes through three given points: one comma two, two comma five, and five comma four. Let's visualize these points on a coordinate plane and see how we can determine the circle that contains all three points. To find the circle equation, we use the general form: x squared plus y squared plus D x plus E y plus F equals zero. We substitute each of the three points into this equation. For point one comma two, we get D plus two E plus F equals negative five. For point two comma five, we get two D plus five E plus F equals negative twenty nine. For point five comma four, we get five D plus four E plus F equals negative forty one. Now we solve this system of three equations. First, we subtract equation one from equation two to get D plus three E equals negative twenty four. Next, we subtract equation one from equation three to get two D plus E equals negative eighteen. From this second result, we can express E as negative eighteen minus two D. Substituting this into our first result and solving, we find D equals negative six. Working backwards, we get E equals negative six and F equals thirteen. With D equals negative six, E equals negative six, and F equals thirteen, our circle equation becomes x squared plus y squared minus six x minus six y plus thirteen equals zero. The center coordinates are h equals three and k equals three, giving us center at three comma three. The radius squared equals h squared plus k squared minus F, which is nine plus nine minus thirteen equals five. Therefore, the radius is square root of five, and the diameter is two times square root of five. To summarize our solution: We used the general circle equation and substituted three given points to create a system of linear equations. By solving this system, we found the coefficients D, E, and F. From these, we calculated the center at three comma three and the radius as square root of five. Therefore, the diameter of the circle is two times square root of five.