MHB How Do You Calculate the BC Angle in a Triangle?

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To calculate the BC angle in a triangle, the length of DE is determined to be 6 using Pythagoras' theorem. The angles α and β are defined, with tan(α) equating to 3/4. Consequently, tan(β) is calculated as 1/7 using the tangent subtraction formula. The lengths BD and AB are derived as 8/7 and 40√2/7, respectively, leading to AC being 40√2 and BC calculated as 400/7. The discussion emphasizes the use of geometry for these calculations.
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By Pythagoras, $\overline{DE} = 6$. With angles $\alpha,\;\beta$ as marked in the diagram, $\tan\alpha = \frac34$. Therefore $$\tan\beta = \tan(45^\circ - \alpha) = \frac{1-\frac34}{1 + \frac34} = \frac17.$$ So $\overline{BD} = \frac87$, and (Pythagoras) $\overline{AB} = \dfrac{40\sqrt2}7.$ It follows that $\overline{AC} = 40\sqrt2$, and (Pythagoras again) $\overline{BC} = \dfrac{400}7.$[/sp]
 

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thanks for your participation, your answer is correct !
this time using geometry only ,hope someone can do it
 

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