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# Coordinates

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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
+1365
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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