Question

A right triangle has a hypotenuse of length 10√2 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. A. The hypotenuse of the second triangle has length 7√2 . B. The hypotenuse of the second triangle has length 14√2 . C. The hypotenuse of the second triangle has length 7. D. The hypotenuse of the second triangle has length 14.

Answer 1

To determine the length of the hypotenuse in the second triangle, we start by noting that both triangles are 45-45-90 triangles. In a 45-45-90 triangle, the sides opposite the 45-degree angles are equal, and the hypotenuse is \( l\sqrt{2} \), where \( l \) is the length of either leg.

For the second triangle:

1. The side opposite the 45-degree angle is given as 14. This side is \( l = 14 \).
2. Therefore, according to the properties of a 45-45-90 triangle, the hypotenuse \( h \) can be calculated as follows: \[ h = l\sqrt{2} = 14\sqrt{2}. \]

Now, we can cross-reference this calculation with the answer choices provided:

A. The hypotenuse of the second triangle has length \( 7\sqrt{2} \) — This is incorrect. B. The hypotenuse of the second triangle has length \( 14\sqrt{2} \) — This is correct. C. The hypotenuse of the second triangle has length \( 7 \) — This is incorrect. D. The hypotenuse of the second triangle has length \( 14 \) — This is incorrect.

Thus, the length of the hypotenuse in the second triangle is \( \boxed{14\sqrt{2}} \).