Final answer:
The equation of the bisector of the angle between two lines, follow these steps: Find the slopes of the lines, calculate the angles of inclination, determine the slope of the bisector using the formula m = tan(theta/2), and use the point-slope form of a line to write the equation of the bisector.
Step-by-step explanation:
To find the equation of the bisector of the angle between two lines, you can follow these steps:
- Find the slopes of the two lines.
- Use the formula m = tan(theta) to find the angles of the lines with the positive x-axis.
- Use the formula m = tan(theta/2) to find the slope of the bisector.
- Use the point-slope form of a line, y - y1 = m(x - x1), to write the equation of the bisector.
For the given lines 3x - 4 - 7 = 0 and 5x - 2y - 8 = 0, the slopes are m1 = -3/5 and m2 = 5/2. The angles with the positive x-axis are theta1 = arctan(-3/5) and theta2 = arctan(5/2).
Using the formula m = tan(theta/2), we find m = tan((theta1 + theta2)/2) = tan((-3.06 + 1.19)/2) = tan(-0.93/2) = -1.04.
Using the point-slope form with one of the points on the bisector (such as the intersection point of the two lines), we can write the equation of the bisector as y - y1 = m(x - x1).