a right triangle has a hypotenuse lenght of 16 and an angle of 45, with a side opposite this angleof lenght 8square root 2. a second right triangle also has an angle of 45 and a side opposite this angle with a lenght of 4 square root 2. determine the lenght of the hypotenuse in the second triangle.

Question

the hypotenuse of the second triangle has lenght 8.
the hypotenuse of the second triangle has lenght 4.
the second hypotenuse has lenght 4 square root 2
the second triangle has lenght 8 square root 2

Answer 1

In a right triangle with a 45-degree angle, the two legs are of equal length, and the relationship between the length of a leg and the hypotenuse \( c \) is given by:

\[ c = \text{leg} \times \sqrt{2} \]

For the second triangle, we are given that the side opposite the 45-degree angle is \( 4\sqrt{2} \). Since this is a 45-45-90 triangle, both legs are equal. Therefore, both legs have a length of \( 4\sqrt{2} \).

Now, using the formula for the hypotenuse:

\[ c = 4\sqrt{2} \times \sqrt{2} = 4 \times 2 = 8 \]

Thus, the length of the hypotenuse in the second triangle is 8.

Therefore, the correct answer is: the hypotenuse of the second triangle has length 8.