Find the circumcenter of angle EFG with E(4,4), F(4,2), and G (8,2).
find the circumcenter of triangle efg with e(4 4) f(4 2) and g(8 2)
To find the circumcenter of angle EFG, you will need to follow these steps:
1. Calculate the midpoints of the two sides of the angle: EF and EG.
- The midpoint of a line segment can be found by averaging the coordinates of its endpoints.
- The midpoint of EF is [(x1 + x2)/2, (y1 + y2)/2].
Given points E(4,4) and F(4,2), the midpoint of EF is [(4 + 4)/2, (4 + 2)/2] = (4, 3).
- The midpoint of EG is [(x1 + x2)/2, (y1 + y2)/2].
Given points E(4,4) and G(8,2), the midpoint of EG is [(4 + 8)/2, (4 + 2)/2] = (6, 3).
2. Calculate the slope of the lines perpendicular to EF and EG.
- To find the slope of a line perpendicular to another line, you need to take the negative reciprocal of the slope.
- The slope of EF is (y2 - y1) / (x2 - x1) = (2 - 4) / (4 - 4) = -2/0 = undefined.
- The slope of the line perpendicular to EF is 0, as the line is vertical.
- The slope of EG is (2 - 4) / (8 - 4) = -2/4 = -1/2.
- The slope of the line perpendicular to EG is 2, as it is the negative reciprocal.
3. Find the equations of the lines passing through the midpoints with the perpendicular slopes.
- The equation of a vertical line is x = k, where k is the x-coordinate of any point on the line.
Therefore, the equation of the line passing through the midpoint of EF is x = 4.
- The equation of a line with slope m passing through a point (x1, y1) is y - y1 = m(x - x1).
Using the midpoint (6, 3) and slope 2, the equation of the line passing through the midpoint of EG is y - 3 = 2(x - 6).
4. Solve the system of equations formed by the two lines.
- The system of equations is x = 4 and y - 3 = 2(x - 6).
Solving for x in the first equation gives x = 4.
Substituting x = 4 into the second equation gives y - 3 = 2(4 - 6) => y - 3 = -4 => y = -1.
Therefore, the intersection point of the two lines is (4, -1), which represents the circumcenter of angle EFG.
So, the circumcenter of angle EFG is at coordinates (4, -1).
Take a look at
http://www.jiskha.com/display.cgi?id=1394471353
and plug in your points instead