Question

1. john is building a triangular treehouse and he knows one side of the triangle. he needs to find the length of the missing side to complete the treehouse. if he rounds his measurements to the nearest tenth, what could be the length of the missing side? 17 16.5 16.7 16.9

Explanation:

Step1: Identify trigonometric ratio

The triangle is right-angled. The known side (12) is adjacent to the 54° angle, and \(x\) is opposite. Use tangent: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

Step2: Set up equation

\(\tan(54^\circ) = \frac{x}{12}\)

Step3: Solve for \(x\)

\(x = 12 \times \tan(54^\circ)\)

Step4: Calculate value

\(\tan(54^\circ) \approx 1.3764\), so \(x \approx 12 \times 1.3764 = 16.5168\), rounded to nearest tenth is 16.5? Wait correction: Wait no, wait if 12 is adjacent, opposite is x, tan(54)=x/12 → x=12tan54≈121.37638≈16.5166≈16.5? Wait but wait maybe 12 is opposite? Wait no, the angle is 54°, let's check again. Wait maybe the 12 is the side adjacent, yes. Wait but the options have 16.5 as an option. Wait wait let me recalculate: tan(54) is approximately 1.37638192047. 121.37638≈16.51656≈16.5 when rounded to nearest tenth. But wait wait another possibility: if 12 is the opposite side, then cot(54)=adjacent/opposite → adjacent=12cot(54)=12/tan54≈12/1.376≈8.72 which is not an option. So the correct answer is 16.5? Wait but wait the options include 16.5, 16.7,16.9,17. Wait let me check again: Oh wait maybe I made a mistake in the angle. Wait 54 degrees: tan(54)≈1.376, so 12*1.376=16.512≈16.5. So the answer is 16.5.

Wait correction: Wait the original problem's image shows a right triangle with angle 54°, one side 12 (adjacent to 54°), missing side x (opposite). So yes, x=12*tan54≈16.5. So the answer is 16.5.

Step1: Choose trigonometric ratio

Use tangent: \(\tan(54^\circ) = \frac{x}{12}\)

Step2: Solve for x

\(x = 12 \times \tan(54^\circ)\)

Step3: Compute value

\(\tan(54^\circ) \approx 1.376\), so \(x \approx 12 \times 1.376 = 16.512\)

Step4: Round to nearest tenth

\(16.512 \approx 16.5\)

Answer:

16.9