A right triangle has a hypotenuse of length 10 square root of 2 and an angle of 45 degrees, with a side opposite this angle with a length if 10. A second right triangle also gas 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.

Question

GPT-4o mini · Answer

To solve the problem, we first need to recall the properties of a right triangle with a 45-degree angle. A right triangle with both angles measuring 45 degrees is an isosceles right triangle, which means that the lengths of the two legs are equal, and they can be expressed in terms of the hypotenuse.

The relationship for an isosceles right triangle (45-45-90 triangle) can be summarized as:

The length of each leg \( a \) relates to the hypotenuse \( c \) by the formula: \[ c = a \sqrt{2} \]

In this context, we can analyze the second triangle which has a leg (side opposite the 45-degree angle) with length 14. This means: \[ a = 14 \]

Using the formula for the hypotenuse: \[ c = 14 \sqrt{2} \]

Thus, the length of the hypotenuse in the second triangle is: \[ \boxed{14\sqrt{2}} \]