A 45-45-90 triangle is a special type of right triangle where the two non-hypotenuse sides (the legs) are of equal length, and the angles are 45 degrees, 45 degrees, and 90 degrees.
To determine the true statement about the relationship between the legs and the hypotenuse in a 45-45-90 triangle, let's examine the properties of such a triangle:
1. In a 45-45-90 triangle, the legs are of equal length.
2. The hypotenuse is longer than either leg and specifically, one of the main properties of such an isosceles right triangle is that the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of either leg.
This geometric property can be observed through the Pythagorean theorem. If we denote the length of each leg as [tex]\(a\)[/tex], then the hypotenuse [tex]\(c\)[/tex] can be calculated as follows:
[tex]\[ c = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \][/tex]
Thus, the hypotenuse is [tex]\(a\sqrt{2}\)[/tex] times the length of either leg.
Let's evaluate each statement:
A. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg.
- False. The hypotenuse is not [tex]\(\sqrt{3}\)[/tex] times the length of either leg in a 45-45-90 triangle.
B. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
- True. This accurately describes the relationship in a 45-45-90 triangle.
C. Each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse.
- False. The legs are not [tex]\(\sqrt{3}\)[/tex] times the length of the hypotenuse.
D. Each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse.
- False. Each leg is [tex]\(1/\sqrt{2}\)[/tex] or [tex]\(\sqrt{2}/2\)[/tex] times the hypotenuse, not [tex]\(\sqrt{2}\)[/tex] times the hypotenuse.
Therefore, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
