MHB Calculating the Length of an Angle Bisector: Is My Solution Correct?

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To calculate the length of the angle bisector $$BK$$ in triangle $$A(1;4), B(7;8), C(9;2)$$, the angle bisector theorem states that the ratio $$\frac{AK}{KC}$$ equals $$\frac{AB}{AC}$$. The lengths of sides $$AB$$ and $$AC$$ can be determined using the distance formula. The equation provided appears to represent the line containing $$BK$$, but verification is needed. Stewart's theorem can also be applied to find the length of $$BK$$ accurately. The discussion emphasizes the importance of using established geometric theorems for verification.
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How to find the length of an angle bisector ($$BK$$) in a triangle $$A(1;4), B(7;8), C(9;2)$$.

I calc $$BK$$:
[math]\frac{x-7}{\frac{1+\frac{\sqrt{13}}{\sqrt{10}} \cdot 9}{1+\frac{\sqrt{13}}{\sqrt{10}}} - 7} - \frac{y-8}{\frac{4+\frac{\sqrt{13}}{\sqrt{10}} \cdot 2}{1+\frac{\sqrt{13}}{\sqrt{10}}} - 8}=0[/math]

Is it right?
 
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Welcome, divisor! (Wave)

Is the equation you wrote supposed to be the equation of line containing $BK$? I haven't checked it for correctness, but you usually find lengths in the coordinate plane using the distance formula. Since $BK$ is an angle bisector of $\triangle ABC$, the angle bisector theorem gives $\frac{AK}{KC} = \frac{AB}{AC}$. Find $AB$ and $AC$ using the distance formula. Next, use the equation in Stewart's theorem to solve for $BK$.
 
Euge, thank you. I want to check my solution by Stewart's theorem.
 
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