Question

A right triangle has a hypotenuse of length 102–√

and an angle of 45 degrees, with a side opposite this angle with a length of 10. A second right triangle also has an angle of 45 degrees, with a side opposite this angle with a length of 14. Determine the length of the hypotenuse in the second triangle.

Answer 1

[AI-generated answer — Bot GPT 3.5]

In the first triangle, the hypotenuse has a length of $102-\sqrt{101}$ and the side opposite the $45^\circ$ angle has a length of 10. Let $x$ be the length of the side adjacent to the $45^\circ$ angle. Then by the Pythagorean Theorem, \begin{align*}

x^2 + 10^2 &= (102-\sqrt{101})^2\\
x^2 + 100 &= 102^2 - 204\sqrt{101} + 101\\
x^2 &= 101 - 204\sqrt{101} + 100\\
x &= \sqrt{101}-\sqrt{100} = \sqrt{101} - 10.
\end{align*} Therefore, the length of the hypotenuse in the second triangle is \begin{align*}
(\sqrt{101} - 10) + 14 &= \sqrt{101} + 4.
\end{align*} The length of the hypotenuse in the second triangle is $\boxed{\sqrt{101}+4}$.

Answer 2

it's easy to get this wrong, since the user does not seem to know how to type math. If the triangles have two angles of 45°, then if one side is 10, the hypotenuse is 10√2

If the other triangle has sides of 14, then its hypotenuse is 14√2

Answer 3

[AI-generated answer — Bot GPT 3.5]

You are correct, I apologize for the confusion. If both triangles have angles of 45 degrees, then the sides are in a ratio of 1:1:√2.

In the first triangle, if one side is 10, then the hypotenuse is $10\sqrt{2}$.

In the second triangle, if one side is 14, then the hypotenuse is $14\sqrt{2}$.

So, the length of the hypotenuse in the second triangle is $\boxed{14\sqrt{2}}$.