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Find the equation of a line that bisects the acute angle formed by the following lines:
2x - y = 3 and x - 2y = 3
Solution:
To illustrate the problem, it is better to draw the graph of the given lines as follows:
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| Photo by Math Principles in Everyday Life |
The two given lines form vertical angles, the acute angles and obtuse angles. Each angles are supplement to each other. There are two angle bisectors in these given lines, the angle bisector for the obtuse angle and the angle bisector for the acute angle. In this problem, we will find the equation of a line that bisects the acute angle. Let's assign a point that is located between the two given lines, and label this as P(x,y). From point P, draw a line that is perpendicular to 2x - y = 3 and label this as d1. From point P, draw a line that is perpendicular to x - 2y = 3 and label this as d2.
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| Photo by Math Principles in Everyday Life |
From the figure above, it shows a distance between a point to a line. The distance between a point to a line is given by the formula
Since we are finding the equation of the angle bisector, we have to equate the distance of a point to the two given lines, which are d1 and d2. Since point P is located below the line 2x - y = 3, then d1 must be negative. Since point P is located above the line x - 2y = 3, then d2 must be positive.
Simplify the above equation
- 2x + y + 3 = x - 2y - 3
3x - 3y - 6 = 0
Divide both sides by 3
x - y - 2 = 0
Therefore, the equation of angle bisector, L is x - y - 2 = 0.