Acellus Learning System ght Triangles Find the indicated side of the triangle. 12 30 b a = [?] Enter sty e A
The Correct Answer and Explanation is:
To solve for the missing side of a right triangle using the Acellus Learning System problem involving angles (likely 30°) and side lengths (12), we’ll assume the following typical setup:
Given:
- A right triangle with one angle = 30°
- The side opposite the 30° angle = 12 units
- We are to find side a (let’s assume it’s either the hypotenuse or the adjacent side based on standard triangle rules)
Step-by-Step Solution:
In a 30°-60°-90° triangle, the side relationships are:
- The side opposite 30° is half the hypotenuse
- The side opposite 60° is √3/2 of the hypotenuse
- The hypotenuse is twice the side opposite 30°
Case 1: If side 12 is opposite the 30° angle
Then the hypotenuse (a) is: a=2×12=24a = 2 \times 12 = 24
So, a = 24 units
Case 2: If side 12 is adjacent to 30° (i.e., between the 30° and 90° angles)
Then side a (opposite 30°) is: tan(30°)=a12⇒a=12×tan(30°)=12×33≈12×0.577=6.93\tan(30°) = \frac{a}{12} \Rightarrow a = 12 \times \tan(30°) = 12 \times \frac{\sqrt{3}}{3} \approx 12 \times 0.577 = 6.93
So, a ≈ 6.93 units
Since the prompt directly connects 12 and 30°, and no triangle diagram is visible, it’s most likely referencing a standard 30-60-90 triangle where 12 is the side opposite 30°.
✅ Final Answer: a = 24 units
Explanation
To find the indicated side of a triangle, we must identify whether the triangle is right-angled and what side and angle we are working with. In this problem, we are told it is a right triangle, and we are given an angle of 30° and a side length of 12.
In a right triangle where one of the angles is 30°, the triangle is called a 30°-60°-90° triangle, which has known side length ratios. Specifically, the side opposite the 30° angle is exactly half the length of the hypotenuse, and the side opposite the 60° angle is the hypotenuse times √3/2.
If the side of 12 is opposite the 30° angle, then we use the rule: Hypotenuse=2×(Opposite 30°)=2×12=24\text{Hypotenuse} = 2 \times (\text{Opposite 30°}) = 2 \times 12 = 24
This means side a, which is the hypotenuse, is 24 units.
If the side of 12 were instead adjacent to the 30° angle, we would use the tangent function, since: tan(30°)=oppositeadjacent⇒opposite=12×tan(30°)\tan(30°) = \frac{\text{opposite}}{\text{adjacent}} \Rightarrow \text{opposite} = 12 \times \tan(30°)
Which gives a smaller value, about 6.93. But since 12 is more commonly the side opposite 30° in textbook-style problems, the correct interpretation is that the hypotenuse is 24.
Understanding these triangle relationships is key in trigonometry and geometry.
