In a 45-45-90 triangle, we have a special type of right triangle where the two legs are of equal length. This triangle is particularly notable because of its consistent angle measures: two angles are 45 degrees each, and the right angle is 90 degrees.
Given this triangle's properties:
1. The legs are congruent, meaning they are of equal length.
2. The relationship between the legs and the hypotenuse follows a specific ratio.
Let's identify the true statement about this sort of triangle from the given options:
A. Each leg is [tex]$\sqrt{3}$[/tex] times as long as the hypotenuse.
- This statement is false. In a 45-45-90 triangle, the hypotenuse is not related to the legs by a factor of [tex]$\sqrt{3}$[/tex].
B. Each leg is [tex]$\sqrt{2}$[/tex] times as long as the hypotenuse.
- This statement is false. If each leg were [tex]$\sqrt{2}$[/tex] times as long as the hypotenuse, the lengths would not form a 45-45-90 triangle.
C. The hypotenuse is [tex]$\sqrt{3}$[/tex] times as long as either leg.
- This statement is false. The factor [tex]$\sqrt{3}$[/tex] does not apply in a 45-45-90 triangle scenario.
D. The hypotenuse is [tex]$\sqrt{2}$[/tex] times as long as either leg.
- This statement is true. In a 45-45-90 triangle, the hypotenuse is always [tex]$\sqrt{2}$[/tex] times the length of either leg. This is a distinctive property of this special kind of triangle.
Therefore, the correct statement about a 45-45-90 triangle is:
The hypotenuse is [tex]$\sqrt{2}$[/tex] times as long as either leg.
Thus, the answer is D.
