To determine the true statement about a 45-45-90 triangle, let's go through the properties of this specific type of triangle.
A 45-45-90 triangle is a special right triangle where both of the legs are of equal length and the angles are [tex]\(45^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(90^\circ\)[/tex].
In a 45-45-90 triangle, the hypotenuse is always [tex]\(\sqrt{2}\)[/tex] times the length of either leg. This is derived from the Pythagorean theorem for a right triangle:
[tex]\[ a^2 + a^2 = c^2 \][/tex]
[tex]\[ 2a^2 = c^2 \][/tex]
[tex]\[ c = a\sqrt{2} \][/tex]
Here, [tex]\(a\)[/tex] represents the length of each leg, and [tex]\(c\)[/tex] represents the length of the hypotenuse.
Given the provided answer, we can conclude that:
- Option A states that each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse. This is incorrect because the hypotenuse is longer than each leg.
- Option B states that the hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg. This is incorrect as it refers to a different type of triangle (30-60-90 triangle).
- Option C states that each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse. This is incorrect as it does not fit any standard right triangle properties.
- Option D correctly states that the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
Thus, the true statement about a 45-45-90 triangle is:
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
