Questions: Find the length of the right triangle's hypotenuse, assuming the shorter leg has length 6 and the longer leg has length 8. True or False? (Select one answer. Using radians as needed.) sin θ = 8/sqrt(100) An approximation of the sine is not needed to answer this question. Simplify your answer (Type an integer or decimal rounded to the nearest hundredth as needed.)

Find the length of the right triangle's hypotenuse, assuming the shorter leg has length 6 and the longer leg has length 8.

True or False? (Select one answer. Using radians as needed.)

sin θ = 8/sqrt(100)

An approximation of the sine is not needed to answer this question.

Simplify your answer (Type an integer or decimal rounded to the nearest hundredth as needed.)

Transcript text: Find the length of the right triangle's hypotenuse, assuming the shorter leg has length 6 and the longer leg has length 8. True or False? (Select one answer. Using radians as needed.) $\sin \theta = \frac{8}{\sqrt{100}}$ An approximation of the sine is not needed to answer this question. Simplify your answer (Type an integer or decimal rounded to the nearest hundredth as needed.)

Solution

Solution Steps

Step 1: Apply the Pythagorean theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, both legs have a length of 6, and we are looking for the length of the hypotenuse, c. So, the equation becomes:

c² = 6² + 6²

Step 2: Simplify the equation

c² = 36 + 36 c² = 72

Step 3: Solve for c

Take the square root of both sides of the equation to solve for c: c = √72

Step 4: Simplify the radical

√72 can be simplified as follows: √72 = √(36 * 2) = √36 * √2 = 6√2

Step 5: Approximate the value

The exact answer is 6√2. To approximate this value, we can use a calculator to find the square root of 2 and multiply it by 6: 6 * √2 ≈ 6 * 1.414 = 8.485 Rounding to the nearest thousandth gives 8.485.