Find the length of side x in simplest radical form with a rational denominator.
30°
60°
4
x
To find the length of side x, we can use the properties of a 30-60-90 triangle. In a 30-60-90 triangle, the lengths of the sides are related by the following ratios:
- The length of the side opposite the 30-degree angle is half the length of the hypotenuse. Let's call this side y.
- The length of the side opposite the 60-degree angle is equal to the length of the hypotenuse multiplied by the square root of 3. This is side x.
- The length of the hypotenuse is twice the length of the side opposite the 30-degree angle. Let's call this side z.
Since we are given that the length of side x is equal to 4, we can use this information to find the length of the other sides.
Using the ratio for side x, we have:
x = z * √3
Substituting the value of z, we have:
4 = (2y) * √3
Dividing both sides by 2 and √3, we get:
4 / (2√3) = y
Simplifying, we get:
2 / √3 = y
To rationalize the denominator, we multiply the numerator and denominator by √3:
(2 / √3) * (√3 / √3) = (2√3) / 3
Therefore, the length of side y is (2√3) / 3.
So, in simplest radical form with a rational denominator, the length of side y is (2√3) / 3.
To find the length of side x in simplest radical form with a rational denominator, we can use the properties of special right triangles. In this case, we have a triangle with angles of 30°, 60°, and 90°.
In a 30°-60°-90° triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is √3 times the length of the side opposite the 30° angle.
We are given that one of the sides is 4 units long. Let's assign this length to the side opposite the 30° angle. Therefore, the side opposite the 60° angle would be 4√3.
Thus, the length of side x, which is opposite the 90° angle, would be twice the length of the side opposite the 30° angle. Therefore, x = 2 * 4 = 8.
So, the length of side x is 8 units.
The answer
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