using slope to find an angle in right angled triangle

Question

I have a right angled triangle in which I know the length and the slope of the hypotenuse, how do I find one of the angles? Thanks

Answer 1

$$\text{Slope} = \dfrac{\text{height}}{\text{base}}=\tan(\ \text{Base}\angle\ )$$

You know the base length and hypotenuse. Then:

$$\text{height} = \sqrt{\text{hypotenuse}^2-\text{base}^2}$$

Putting the value of $height$ in the first equation, we get $$\dfrac{\sqrt{\text{hypotenuse}^2-\text{base}^2}}{\text{base}}=\tan(\ \text{Base}\angle\ )$$

$$\implies\ \text{Base}\ \angle=\arctan\left(\dfrac{\sqrt{\text{hypotenuse}^2-\text{base}^2}}{\text{base}}\right)$$