Question

Special right triangles quick check help please! can someone check if these answers are right?

Answer 1

We have that cos30=$\frac{7\sqrt{3}}{w} = \frac{\sqrt{3}}{2}$ by the definition of cos as the ratio of adjacent leg over the hypotenuse.  Hence, 7/w=1/2 by eradicating square roots. Solving for w in this equation we get that w=7*2=14.
We have that the 3 sides obey the Pythagoran theorem, because:$(7\sqrt{3})^2+21^2=(14\sqrt{3})^2 because 488=147+441$
Hence, the triangle is right. We also have that one side is half the hypotenuse hence one of the other angles of the triangle is 30 degrees because sin30=1/2. Finally, the third angle is 60 degrees because sum of angles=180.
Here, we have that sin 45=p/44=$\frac{\sqrt2}{2}$. Multiplying by 44, we have that p=22$\sqrt{2}$