Question

Consider the given right triangle. If ∠B=30∘ and a=8√3 m, then ∠A=60 and c = ? m.

Answer 1

Using the properties of a 30-60-90 triangle and the Pythagorean theorem, we can find that the hypotenuse c of the triangle is 16 meters long.

Step-by-step explanation:

The question involves using the Pythagorean theorem to find the length of the hypotenuse c in a right triangle where one angle is 30 degrees and the adjacent side to this angle (a) is 8√3 meters. Given that the other angle (∠A) is 60 degrees, by the properties of a 30-60-90 triangle, we know the relationship between the lengths of the sides is such that the hypotenuse (c) is twice the length of the shorter leg (side opposite the 30-degree angle), and the longer leg (side opposite the 60-degree angle) is √3 times the length of the shorter leg.

Here, side a (8√3 m) is opposite the 60-degree angle, so it is the longer leg. Thus, we can find the length of the hypotenuse by:

• Determining the length of the shorter leg (b), which is half of c.
• Using the Pythagorean theorem, c = √(a² + b²), where a is 8√3 m.
• Calculating c once we know b.

Since a = 8√3 m and this is the longer leg of a 30-60-90 triangle, the shorter leg b = a/√3 = (8√3)/√3 = 8 m. Now, the hypotenuse (c) = 2 × (b) = 2 × 8 m = 16 m.