Question

select the correct answer.
what is the value of x in the triangle?
triangle image with right angle, 30°, 60°, hypotenuse? wait, no, the side labeled 15 is adjacent to 30° and the right angle, x is opposite 30°? wait, the triangle has a right angle, 30°, 60°, side 15 (adjacent to 30°), x is the side opposite 30°? wait, no, lets check: angles are 30°, 60°, 90°, so its a 30-60-90 triangle. the side adjacent to 30° is 15? wait, no, in a 30-60-90 triangle, the sides are in ratio 1 : √3 : 2, where the side opposite 30° is the shortest (1), opposite 60° is √3, hypotenuse is 2. wait, the side labeled 15: lets see the angles. the angle at the left is 30°, right angle at the top, so the side with length 15 is adjacent to 30° and opposite to 60°? wait, no, the right angle is at the top, so the horizontal side is 15 (adjacent to 30°), vertical side is x (opposite to 30°), and hypotenuse is the other side. wait, in 30-60-90 triangle, tan(30°) = opposite/adjacent = x/15? wait, no, tan(30°) is 1/√3, so x = 15 tan(30°)? no, wait, maybe i got the sides wrong. wait, the angle at the bottom is 60°, so the side adjacent to 60° is x, and opposite is 15? wait, no, lets re-express: right triangle, angles 30°, 60°, 90°. lets denote: angle 30° (left), angle 60° (bottom), right angle (top). so the sides: the side between 30° and right angle is 15 (lets call this side a, adjacent to 30°), the side between 60° and right angle is x (side b, adjacent to 60°), and the hypotenuse is between 30° and 60°. in a 30-60-90 triangle, the side opposite 30° is the shortest, which is side b (x) if 30° is at left? wait, no, angle at left is 30°, so the side opposite 30° is the side opposite to it, which is x (since 30° is at left, opposite side is x, which is the vertical side). the side adjacent to 30° is 15 (horizontal side). so in 30-60-90 triangle, tan(30°) = opposite/adjacent = x/15. but tan(30°) is 1/√3, so x = 15/√3 = 5√3? wait, no, thats not one of the options? wait, no, maybe i mixed up the angles. wait, the angle at the bottom is 60°, so the side opposite 60° is 15 (horizontal side), and side opposite 30° is x (vertical side). so in 30-60-90 triangle, the side opposite 60° is √3 times the side opposite 30°. so 15 = √3 x → x = 15/√3 = 5√3? wait, but option d is 5√3, option a is 15√3, b is 7.5, c is 5, e is 15. wait, maybe i got the sides wrong. wait, the hypotenuse: in 30-60-90, hypotenuse is twice the shortest side (opposite 30°). so if the side opposite 30° is x, then hypotenuse is 2x. the side opposite 60° is x√3. now, the side labeled 15: is that the side opposite 60°? then x√3 = 15 → x = 15/√3 = 5√3, which is option d. yes, that makes sense. so the ocr text is: select the correct answer. what is the value of x in the triangle? triangle with 30°, 60°, 90°, side 15 (opposite 60°), x (opposite 30°) options: a. 15√3, b. 7.5, c. 5, d. 5√3, e. 15

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here \(\theta = 30^{\circ}\), the adjacent side to \(30^{\circ}\) is \(15\) and the opposite side is \(x\). But we can also use the property of \(30 - 60-90\) triangle. In a \(30 - 60 - 90\) triangle, if the side adjacent to \(30^{\circ}\) (longer leg) is \(l\), and the side opposite to \(30^{\circ}\) (shorter leg) is \(s\), the relationship is \(s=\frac{l}{\sqrt{3}}\). Another way: \(\tan30^{\circ}=\frac{x}{15}\). Since \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\).

Step2: Solve for \(x\)

We have the equation \(\frac{1}{\sqrt{3}}=\frac{x}{15}\). Cross - multiply gives \(x = \frac{15}{\sqrt{3}}\). Rationalize the denominator: \(x=\frac{15\sqrt{3}}{3}=5\sqrt{3}\).

Answer:

D. \(5\sqrt{3}\)