To solve this problem, we need to understand the properties of a 45-45-90 triangle. A 45-45-90 triangle is a special type of isosceles right triangle where the two legs are of equal length, and the hypotenuse is longer than either leg.
The key property of a 45-45-90 triangle is that the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as each of the legs. This can be derived from the Pythagorean Theorem. However, I will directly provide the property for simplicity.
Let's validate each option against this property:
### Option A: Each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse.
- This statement is incorrect. Based on our knowledge of 45-45-90 triangles, each leg is not longer than the hypotenuse by a factor of [tex]\(\sqrt{2}\)[/tex]. Instead, the hypotenuse is longer than the legs by a factor of [tex]\(\sqrt{2}\)[/tex].
### Option B: The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
- This statement is correct. As discussed, the hypotenuse is indeed [tex]\(\sqrt{2}\)[/tex] times as long as either leg in a 45-45-90 triangle. If each leg has length [tex]\(x\)[/tex], the hypotenuse will be [tex]\(x\sqrt{2}\)[/tex].
### Option C: The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg.
- This statement is incorrect. The ratio [tex]\(\sqrt{3}\)[/tex] applies to 30-60-90 triangles, not 45-45-90 triangles. Therefore, this relationship does not hold for 45-45-90 triangles.
### Option D: Each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse.
- This statement is incorrect. Again, the [tex]\(\sqrt{3}\)[/tex] factor pertains to 30-60-90 triangles, not 45-45-90 triangles. The legs in a 45-45-90 triangle are not affected by [tex]\(\sqrt{3}\)[/tex].
Therefore, the true statement about a 45-45-90 triangle is:
B. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
