Final answer:
To find the length of the hypotenuse in a right triangle given an angle and the length of the opposite side, we can use the sine function. With ∠ θ = 53° and the opposite side measuring 2.3 meters, the hypotenuse is approximately 2.9 meters, which is option B.
Step-by-step explanation:
To solve the mathematical problem completely and find the length of the hypotenuse in a right triangle where one angle (∠ θ) is 53° and the side opposite to ∠ θ, also known as a leg of the triangle, is equal to 2.3 meters, we can utilize the Pythagorean theorem. The theorem is defined by the equation a² + b² = c², where 'a' and 'b' are the lengths of the legs of the triangle, and 'c' is the length of the hypotenuse.
To apply this, we first need to identify 'a' as 2.3 meters (the side opposite to the given angle). Since this is a right triangle and we are given one angle (∠ θ) and the opposite side, we can assume that 'a' is opposite to the given angle and 'b' is the adjacent side to that angle. The hypotenuse 'c' will be the longest side of the triangle. So, we need to calculate the value of 'c' using the theorem, which rearranges to c = √(a² + b²).
However, in this particular case, we only know the length of one leg ('a'), and we cannot compute 'c' without knowing 'b'. Therefore, we need to use trigonometric functions such as sine, cosine, or tangent in conjunction with the given angle to find 'b'. Typically, to find the hypotenuse directly with only one side and an angle given, we would use the sine function: θ = sin∑(opposite/hypotenuse), which rearranges to hypotenuse = opposite/sin(θ).
With the given information of θ = 53° and the opposite side (a) being 2.3 meters, the calculation becomes c = 2.3 / sin(53°). After computing this with a calculator, the length of the hypotenuse comes out to be approximately 2.9 meters, which corresponds to option B.