A right triangle has sides of approximate length 3.13, 10.23, and 10.7. What is the angle opposite longest leg? Answer in degrees.
The correct answer and explanation is:
To find the angle opposite the longest leg (which is 10.7 units), we must first identify whether the triangle is valid and then use trigonometry to solve the problem.
Step 1: Identify the Triangle Type
Given side lengths:
- a = 3.13
- b = 10.23
- c = 10.7
We are told it is a right triangle. In a right triangle, the longest side is the hypotenuse.
Let’s test if this triangle satisfies the Pythagorean Theorem: a2+b2=c2(check if it’s a right triangle)a^2 + b^2 = c^2 \quad \text{(check if it’s a right triangle)} 3.132+10.232=9.7969+104.6529≈114.453.13^2 + 10.23^2 = 9.7969 + 104.6529 ≈ 114.45 10.72=114.4910.7^2 = 114.49
The values are very close, so we can confirm it’s a right triangle with sides 3.13, 10.23, and 10.7 (hypotenuse).
Step 2: Use Sine Function
To find the angle opposite the longest leg (10.7) — note that 10.7 is the hypotenuse, so we can’t find an angle opposite to it in a right triangle (the angle opposite the hypotenuse is always 90°).
However, the question seems to ask for the angle opposite the longest leg (not hypotenuse). But 10.7 is not a leg, it’s the hypotenuse. The longest leg here is 10.23, and we’ll find the angle opposite that.
Let’s denote:
- Opposite side = 10.23
- Hypotenuse = 10.7
Use the sine function: sin(θ)=oppositehypotenuse=10.2310.7≈0.9551\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{10.23}{10.7} ≈ 0.9551
Now take the inverse sine: θ=sin−1(0.9551)≈72.83∘\theta = \sin^{-1}(0.9551) ≈ 72.83^\circ
✅ Final Answer: Approximately 72.83°
🔍 Explanation (Summary):
In a right triangle, the angle opposite a side can be found using trigonometric functions. Since the side 10.23 is the longest leg (not the hypotenuse), and the hypotenuse is 10.7, we used the sine function: sin(θ)=opposite leghypotenuse\sin(\theta) = \frac{\text{opposite leg}}{\text{hypotenuse}}
By solving using inverse sine, we found the angle is approximately 72.83 degrees, which is a reasonable value for a right triangle where one angle is already 90°.