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Two black lines pass through the origin, as shown below.  If their slopes are \(2\) and \(4\), then find the slope of the red line that bisects the acute angle between these lines.

 

 Apr 30, 2022
 #1
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The middle slope is 13/4.

 Apr 30, 2022
 #2
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Draw the line \(x = 1\) and let the red line be \(y = kx\)

 

The points of intersection are: \((1,4)\)\((1, 2)\), and \((1, k)\) .

 

Using the Pythagorean Theorem, we find that the lines of the triangle bounded by the 2 lines and x= 1 are \(2\)\(\sqrt 5 \), and \(\sqrt {18}\).

 

Let the distance between \((1,k)\) and \((1,4)\) be \(d\). This means the distance from the points \((1,2)\) and \((1, k)\) is \(2 - d\).

 

Using the Angle Bisector Theorem, we can form the following equation: \({ d \over \sqrt{18}} = {2 - d \over \sqrt{5}}\)

 

Now, we have to solve for d and subtract that from 4.

 

Can you take it from here? 

 May 3, 2022

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