Lesson Finding the slope of the bisector to the angle formed by two given lines in a coordinate plane

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Finding the slope of the bisector to the angle formed by two given lines in a coordinate plane


Problem 1

An acute angle is formed by two lines of slopes  (1/2)  and  (2/11).
What is the slope of the line which bisects the angle?

Solution 

The given slopes are the tangent values of corresponding angles.


Thus we have  tan(a) = 1%2F2  for one angle, "a", and  tan(b) = 2%2F11  for the other angle, "b".


They ask you about  tan%28%28a%2Bb%29%2F2%29.


Using well known formulas of Trigonometry,


    tan%28a%2Bb%29 = %28tan%28a%29+%2B+tan%28b%29%29%2F%281+-+tan%28a%29%2Atan%28b%29%29 = %28%281%2F2+%2B+2%2F11%29%29%2F%281+-+%281%2F2%29%2A%282%2F11%29%29 = %28%2811%2F22+%2B+4%2F22%29%29%2F%28%281+-+1%2F11%29%29 = %28%2815%2F22%29%29%2F%2810%2F11%29%29 = %2815%2A11%29%2F%2810%2A22%29 = 15%2F20 = 3%2F4.


The last step is to use the formula for  tan(c/2) via  tan(c)


    tan%28c%2F2%29 = %28-1+%2B-+sqrt%281%2Btan%5E2%28c%29%29%29%2Ftan%28c%29.


When you apply it, you will get the ANSWER 


    tan%28%28a%2Bb%29%2F2%29 = %28%28-1+%2B+sqrt%281+%2B+%283%2F4%29%5E2%29%29%29%2F%283%2F4%29%29 = %28-1+%2B+sqrt%2825%2F16%29%29%2F%283%2F4%29 = %28-1+%2B+5%2F4%29%2F%283%2F4%29 = %28%281%2F4%29%29%2F%28%283%2F4%29%29 = 1%2F3.


According to the condition, you may use the sign " + " at sqrt instead of " +/- ".


ANSWER.  The slope is  1%2F3.


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