A right triangle has a hypotenuse of length 16 and an angle of 45°, with a side opposite this angle of length 82–√

. A second right triangle also has an angle of 45° and a side opposite this angle with a length of 42–√
. Determine the length of the hypotenuse in the second triangle.(1 point)
Responses

The hypotenuse of the second triangle has length 42–√
.
The hypotenuse of the second triangle has length 4 Start Root 2 End Root .

The hypotenuse of the second triangle has length 4.
The hypotenuse of the second triangle has length 4.

The hypotenuse of the second triangle has length 8.
The hypotenuse of the second triangle has length 8.

The hypotenuse of the second triangle has length 82–√
.

1 answer

In a right triangle with a 45° angle, the lengths of the two legs are equal, and if we let \( x \) represent the length of each leg, then the hypotenuse \( h \) can be calculated using the Pythagorean theorem:

\[ h = x\sqrt{2} \]

For the second triangle, we know the length of the side opposite the 45° angle is \( \sqrt{42} \). Therefore, both legs of this triangle are \( \sqrt{42} \), and we can find the hypotenuse as follows:

\[ h = \sqrt{42}\sqrt{2} = \sqrt{42 \cdot 2} = \sqrt{84} \]

Next, we can simplify \( \sqrt{84} \):

\[ \sqrt{84} = \sqrt{4 \cdot 21} = 2\sqrt{21} \]

Thus, the length of the hypotenuse in the second triangle is \( 2\sqrt{21} \).

However, it seems that none of the provided responses match this result. The closest we can identify from the available options based on the construction of the triangles under standard conditions is that the hypotenuse of the second triangle does not correspond to any of the responses. Please double-check if there is a specific context or condition you may want to mention additionally for clarification.