Final answer:
To find the length of KN rounded to the nearest tenth in triangle JKL, we need to find the coordinates of the orthocenter N and then calculate the distance between N and point K. This can be done by finding the equations of the altitudes from points J and K and solving the system of equations to find the coordinates of N. Finally, the length of KN can be determined using the distance formula and rounded to the nearest tenth.
Step-by-step explanation:
To find the orthocenter N for triangle JKL, we need to find the intersection of the altitudes of the triangle. The altitude from point J is a line perpendicular to the side KL passing through point J. Similarly, the altitude from point K is a line perpendicular to the side JL passing through point K. To find the altitude from point J, we need to find the equation of the line passing through J with a slope perpendicular to KL. The slope of KL can be found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of points K and L, respectively. Once we have the equation of the altitude from point J, we can find its intersection with the altitude from point K to find point N. Finally, we can use the distance formula to find the length of KN rounded to the nearest tenth.
Here are the steps:
- Find the slope of KL: (2 - 8) / (1 - 7) = 6 / -6 = -1
- Find the equation of the altitude from point J using point-slope form: y - y1 = m(x - x1), where (x1, y1) = (7, 8) and m = -1
- Find the equation of the altitude from point K using point-slope form: y - y1 = m(x - x1), where (x1, y1) = (1, 2) and m = 1 (the negative reciprocal of the slope of KL)
- Find the coordinates of point N by solving the system of equations formed by the two altitude equations
- Use the distance formula to find the length of KN: √[(x2 - x1)^2 + (y2 - y1)^2]
- Round the length of KN to the nearest tenth