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# Angle bisector from two lines of graph

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We have the lines y=3x and y=2x. Let y=mx be the angle bisector of these two lines. How can I find "m"?

Jun 21, 2024

#1
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To find the slope $$m$$ of the angle bisector of the lines $$y = 3x$$ and $$y = 2x$$, we can use the formula for the slope of the angle bisector between two lines given their slopes $$m_1$$ and $$m_2$$:

Given the lines $$y = m_1 x$$ and $$y = m_2 x$$, the slope $$m$$ of the angle bisector is given by:

$m = \frac{m_1 + m_2}{1 + m_1 m_2}$

Here, $$m_1 = 3$$ and $$m_2 = 2$$. Plugging these values into the formula, we get:

$m = \frac{3 + 2}{1 + 3 \cdot 2} = \frac{5}{1 + 6} = \frac{5}{7}$

Thus, the slope $$m$$ of the angle bisector is $$\frac{5}{7}$$.

Jun 21, 2024
#2
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Correct me if I'm wrong but looking around, I saw the formula

(A1x + B1y + C1)/√(A1^2 + B12) = + (A2 x+ B2y + C2)/√(A2^2 + B2^2)

with

L1 : A1x + B1y + C1 = 0

L2 : A2x + B2y + C2 = 0 .

Does this do the same thing? or is it a different formula

#3
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There is a formula similar to the one you showed.

The equation states that we have $$\frac{Ax+By+C}{\sqrt{A^2+B^2}} = \pm \frac{ax+by+c}{\sqrt{a^2+b^2}}$$ for two lines in the form of $$Ax + By + C =0, ax + by + c =0$$

It's similar, but the two lines given are in standard form rather than slope-intercept form.

I'm not too familiar on this equation, but I hope this clarified this a bit.

Feel free to ask if you're still confused.

Thanks! :)

Jun 21, 2024
edited by NotThatSmart  Jun 21, 2024
#4
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Is the formula used for calculating an angle bisector?

#5
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Unfortunately, I don't think it's used to calculate the angle bisector of a line.

It is mainly used to calculate if two lines are parrallel to eachother.

Since $$\frac{Ax+By+C}{\sqrt{A^2+B^2}}$$ calculates the distance from the origin for the line $$Ax + By + C =0$$ and the same applies for the left hand side, if the two are equal, they are parrallel lines.

However, I do believe I know what formula was used.

If we set the slopes of the two lines given in the question to $$m_1$$ and $$m_2$$, the slope of the angle bisector is

$$m=\sqrt{\frac{m_1m_2+1}{m_1+m_2}}$$

I hope this helps! If you have anymore questions, feel free to ask!

Thanks! :)

NotThatSmart  Jun 21, 2024
edited by NotThatSmart  Jun 21, 2024
edited by NotThatSmart  Jun 21, 2024
#7
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Finally got it. The formula you entered was actually correct and the answer was y=(sqrt2 + 1)x. I watched a video and it explained it quite well.

#9
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I'm not sure if  $$y=(\sqrt2 + 1)x$$ is correct though.

Through my understanding, the answer must be $$y=\sqrt{\frac{7}{5}}x$$ is the full equation of the line.

Am I mistaken with this judgement?

I entered the problem with an AI math bot...it confirmed my answer...

Thanks! :)

NotThatSmart  Jun 21, 2024
edited by NotThatSmart  Jun 21, 2024
#6
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I'll tackle this problem.

First, we have two lines given, $$y=3x$$ and $$y=2x$$

Let's note the slopes of the two line. The slope for the first line is 3, and the slope for the second line is 2.

Let's set $$m_1 = 3$$ and $$m_2 = 2$$

Using the angle bisector formula, which states that we have

$$m=\sqrt{\frac{m_1m_2+1}{m_1+m_2}}$$ where m1 and m2 are the slopes of the two lines, we have

$$m = \pm \sqrt{\frac{3*2+1}{3+2}}\\ m = \pm \sqrt{\frac{7}{5}}$$

So our final answer is $$m = \pm \sqrt{\frac{7}{5}}$$

Thanks! :)

Jun 21, 2024