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.(1 point) Responses The hypotenuse of the second triangle has length 14. The hypotenuse of the second triangle has length 14. The hypotenuse of the second triangle has length 7. The hypotenuse of the second triangle has length 7. The hypotenuse of the second triangle has length 142–√ . The hypotenuse of the second triangle has length 14 Start Root 2 End Root . The hypotenuse of the second triangle has length 72–√ . The hypotenuse of the second triangle has length 7 Start Root 2 End Root .

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GPT-4o mini · Answer

In a right triangle with a $45^\circ$ angle, the two legs are equal in length, and the hypotenuse can be calculated using the formula: $$ \text{Hypotenuse} = \text{Leg length} \cdot \sqrt{2} $$ For the second triangle, the leg opposite the $45^\circ$ angle has a length of $14$. Thus, the hypotenuse can be calculated as follows: $$ \text{Hypotenuse} = 14 \cdot \sqrt{2} $$ The hypotenuse of the second triangle is therefore $14\sqrt{2}$. Now looking at the options provided: - The hypotenuse of the second triangle has length $14$. - The hypotenuse of the second triangle has length $7$. - The hypotenuse of the second triangle has length $14\sqrt{2}$. - The hypotenuse of the second triangle has length $7\sqrt{2}$. - The hypotenuse of the second triangle has length $\sqrt{72}$. The correct choice is: **The hypotenuse of the second triangle has length $14\sqrt{2}$.**