Question

What is the length of side x in a 30-60-90 triangle where one side is the square root of 3 and the other side is unknown?

Answers

  1. 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:

    x^{2} = a^{2} + b^{2}    (1)

    So, we need to find the side “b”. We can calculate it with the given angles.

    From the side “a” we have:

    cos(\alpha) = \frac{a}{x}

    cos(60) = \frac{\sqrt{3}}{x}    (2)

    From the side “b”:

    sin(\alpha) = \frac{b}{x}

    sin(60) = \frac{b}{x}   (3)

    Now, we can calculate “b” by dividing equation (3) by equation (2).

    tan(60) = \frac{\frac{b}{x}}{\frac{\sqrt{3}}{x}}

    b = tan(60)*\sqrt{3} = 3

    Finally, we can find the length of the hypotenuse with equation (1):

    x = \sqrt{a^{2} + b^{2}} = \sqrt{(\sqrt{3})^{2} + (3)^{2}} = 2\sqrt{3}

    Therefore, the length of side x in a 30-60-90 triangle is 2√3.

    I hope it helps you!

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