Given the points K=(-2,1) and m(3,4), find coordinates for a point J that makes angle JKM a right angle.
To find the coordinates for point J that makes angle JKM a right angle, we need to determine the equation of the perpendicular bisector of the line segment KM. The point J will then lie on this perpendicular bisector. Here's how you can find the coordinates for point J:
1. Find the midpoint of the line segment KM. The midpoint is the average of the x-coordinates and the average of the y-coordinates of points K and M. Let's call the midpoint (x_m, y_m).
x_m = (x_K + x_M)/2 = (-2 + 3)/2 = 1/2 = 0.5
y_m = (y_K + y_M)/2 = (1 + 4)/2 = 5/2 = 2.5
Therefore, the midpoint of KM is (0.5, 2.5).
2. Find the slope of the line segment KM. The slope between two points (x1, y1) and (x2, y2) is given by (y2 - y1)/(x2 - x1). Let's call the slope of KM m.
m = (y_M - y_K)/(x_M - x_K) = (4 - 1)/(3 - (-2)) = 3/5 = 0.6
3. The slope of the perpendicular bisector is the negative reciprocal of the slope of KM. Let's call the slope of the perpendicular bisector m_perpendicular.
m_perpendicular = -1/m = -1/0.6 = -5/3 ≈ -1.67
4. Use the coordinates of the midpoint (x_m, y_m) and the slope of the perpendicular bisector (m_perpendicular) to find the equation of the line. The equation of a line in slope-intercept form (y = mx + b) can be written as y - y1 = m(x - x1), where (x1, y1) is a point on the line.
y - 2.5 = -5/3(x - 0.5)
y - 2.5 = -5/3x + 5/6
y = -5/3x + 5/6 + 2.5
y = -5/3x + 5/6 + 15/6
y = -5/3x + 20/6
Simplifying the fraction:
y = -5/3x + 10/3
Therefore, the equation of the perpendicular bisector of KM is y = -5/3x + 10/3.
5. Now, to find the coordinates of point J, we need to solve the equations of the perpendicular bisector and the line KM.
Substitute the coordinates of point K (-2, 1) into the equation of the perpendicular bisector:
y = -5/3x + 10/3
1 = -5/3(-2) + 10/3
1 = 10/3 + 10/3
1 = 20/3
3 = 20 (Invalid equation)
From this, we can see that there is no intersection between the line KM and the perpendicular bisector. Therefore, there is no point J that forms a right angle with line segment KM.