Answer:
The length of side x in a 30-60-90 triangle is 2√3.
Step-by-step explanation:
The numbers 30-60-90 are angles, so we need to find the side x of a right triangle with the following information:
θ: is one angle of the right triangle = 30°
α: is the other angle of the right triangle = 60°
a: is one side of the right triangle = √3
b: is the other side of the right triangle =?
x: is the hypotenuse of the right triangle =?
The length of the hypotenuse can be found by Pitagoras:
[tex] x^{2} = a^{2} + b^{2} [/tex] (1)
So, we need to find the side "b". We can calculate it with the given angles.
From the side "a" we have:
[tex] cos(\alpha) = \frac{a}{x} [/tex]
[tex] cos(60) = \frac{\sqrt{3}}{x} [/tex] (2)
From the side "b":
[tex] sin(\alpha) = \frac{b}{x} [/tex]
[tex] sin(60) = \frac{b}{x} [/tex] (3)
Now, we can calculate "b" by dividing equation (3) by equation (2).
[tex] tan(60) = \frac{\frac{b}{x}}{\frac{\sqrt{3}}{x}} [/tex]
[tex] b = tan(60)*\sqrt{3} = 3 [/tex]
Finally, we can find the length of the hypotenuse with equation (1):
[tex] x = \sqrt{a^{2} + b^{2}} = \sqrt{(\sqrt{3})^{2} + (3)^{2}} = 2\sqrt{3} [/tex]
Therefore, the length of side x in a 30-60-90 triangle is 2√3.
I hope it helps you!
