How do you find the remaining side of a 30^circ-60^circ-90^circ306090 triangle if the side opposite 60^circ60 is 4?

1 Answer
Aug 6, 2017

(4sqrt3)/3433 and (8sqrt3)/3833

Explanation:

A 30^@ - 60^@ - 90^@306090 triangle is a type of special right triangle because the side lengths always occur in the ratio x - x sqrt3 - 2xxx32x.

The 30^@30 angle is opposite to the side length xx, the 60^@60 angle is opposite to xsqrt3x3, and the 90^@90 angle is opposite to 2x2x, as shown below.

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If the side opposite to 60^@60 is 44, then xsqrt 3 = 4x3=4. To find the remaining sides, first solve for xx:

xsqrt 3 = 4x3=4

x = 4/sqrt3x=43 -> divide both sides by sqrt33

x = 4/sqrt3 * color(red)(sqrt3/sqrt3)x=4333 -> rationalize the denominator by multiplying by sqrt3/sqrt333, which is equal to 11

x = (4sqrt3)/3x=433 -> multiply

Lastly, to find the side opposite to the 90^@90 angle, find 2x2x:

2x = 2 * (4sqrt3)/3 = (8sqrt3)/32x=2433=833

So, the sides are (4sqrt3)/3, 4,433,4, and (8sqrt3)/3833.