Question

find the length of side x to the nearest tenth.
(right triangle with angles 30°, 60°, right angle; one leg is 4, the other leg (opposite 30°) is x? wait, no, the right angle is at the bottom right, angle at top is 60°, angle at bottom left is 30°, the leg adjacent to 30° is 4? wait, the leg labeled x is adjacent to 60°? wait, the triangle: right angle at bottom right, so the sides: horizontal leg (adjacent to 30°) is 4, vertical leg (opposite 30°) is x? wait, no, angle at bottom left is 30°, so the side opposite 30° is x? wait, the triangle has angles 30°, 60°, 90°, so its a 30-60-90 triangle. in a 30-60-90 triangle, the sides are in ratio 1 : √3 : 2, where the side opposite 30° is the shortest (length a), opposite 60° is a√3, hypotenuse is 2a. wait, in the diagram, the horizontal leg (adjacent to 30°) is 4? wait, no, the vertical leg is x, horizontal leg is 4. wait, angle at bottom left is 30°, so the side opposite 30° is x (vertical leg), and the side adjacent to 30° is 4 (horizontal leg). wait, no, in a right triangle, tan(30°) = opposite/adjacent = x/4? wait, no, angle 30°: opposite side is x, adjacent side is 4? wait, no, the right angle is at bottom right, so the two legs: horizontal (bottom) is 4, vertical (right) is x. the angle at bottom left is 30°, so the angle between the horizontal leg and the hypotenuse is 30°, so the opposite side to 30° is x (vertical leg), adjacent is 4 (horizontal leg). so tan(30°) = x/4? wait, no, tan(theta) = opposite/adjacent. theta is 30°, opposite is x, adjacent is 4. so x = 4 tan(30°)? wait, no, wait, 30-60-90 triangle: the side opposite 30° is the shortest, so if the hypotenuse is c, then side opposite 30° is c/2, side opposite 60° is (c/2)√3. wait, maybe the horizontal leg (length 4) is adjacent to 30°, so its the side opposite 60°, which is a√3, where a is the side opposite 30° (x). so 4 = x√3, so x = 4 / √3 ≈ 2.3? wait, no, maybe i got the sides reversed. wait, angle at bottom left is 30°, so the side opposite 30° is x (vertical leg), and the side adjacent to 30° is 4 (horizontal leg). wait, no, in a right triangle, the side opposite 30° is the shortest. so if the horizontal leg is 4, and its adjacent to 30°, then the hypotenuse is 4 / cos(30°), and the opposite side (x) is 4 tan(30°). wait, tan(30°) is 1/√3 ≈ 0.577, so 4 (1/√3) ≈ 2.3. but maybe the horizontal leg is the longer leg (opposite 60°), so in 30-60-90, longer leg (opposite 60°) is a√3, shorter leg (opposite 30°) is a. so if longer leg is 4, then shorter leg (x) is 4 / √3 ≈ 2.3. alternatively, if the horizontal leg is the shorter leg (a = 4), then longer leg (x) is 4√3 ≈ 6.9. wait, the diagram: angle at bottom left is 30°, so the side opposite 30° is x (vertical leg), and the side adjacent to 30° is 4 (horizontal leg). wait, no, the right angle is at bottom right, so the vertical leg is x (right side), horizontal leg is 4 (bottom side). so angle at bottom left (30°) is between the bottom leg (4) and the hypotenuse. so the opposite side to 30° is x (vertical leg), adjacent is 4 (horizontal leg). so tan(30°) = x / 4 → x = 4 tan(30°) ≈ 4 * 0.577 ≈ 2.3. but maybe i mixed up. wait, 30-60-90 triangle: sides are in ratio 1 : √3 : 2, where 1 is opposite 30°, √3 opposite 60°, 2 hypotenuse. so if the side opposite 60° is 4, then the side opposite 30° (x) is 4 / √3 ≈ 2.3. if the side opposite 30° is 4, then side opposite 60° is 4√3 ≈ 6.9. wait, the diagram: the vertical leg is x, horizontal leg is 4. the angle at top is 60°, so angle at top is between hypotenuse and vertical leg (x), so that angle is 60°, so the side opposite 60° is the horizontal leg (4), so 4 = x√3 (since in 30-60-90, side opposite 60° is a√3, where a is side opposite 30° (x)). so x = 4 / √3 ≈ 2.3. so the problem is to find x in a 30-60-90 triangle, using trigonometry or the 30-60-90 ratios.

Explanation:

Step1: Identify trigonometric ratio

For the $30^\circ$ angle, side $x$ is opposite, and the given side (length 4) is adjacent. Use tangent: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$

Step2: Substitute known values

$\tan(30^\circ)=\frac{x}{4}$

Step3: Solve for $x$

$x=4\times\tan(30^\circ)$
$\tan(30^\circ)=\frac{\sqrt{3}}{3}\approx0.577$, so $x\approx4\times0.577$

Step4: Calculate and round

$x\approx2.308$, rounded to nearest tenth is $2.3$

Answer:

$2.3$